Operators on Banach spaces of Dirichlet series and semigroups of analytic functions

Programa de Doctorado “Matemáticas” PhD Dissertation Operators on Banach spaces of Dirichlet series and semigroups of analytic functions Author Carlos Gómez Cabello Supervisors Prof. Manuel D. Contreras Márquez Prof. Luis Rodríguez Piazza Sevilla, 2024 A mis padres, Manolo y María José. A mi hermano, Javi. A Isabel. Agradecimientos Y pasa la vida y no has notado que has vivido cuando Y pasa la vida, pasa la vida. Pata Negra Esta es la parte de la tesis dedicada al público general. Escrita desde el cariño. Es para vosotros. Lo que viene después, quizá no tanto. Lo mío con Sevilla dura ya unos años. En el momento de escribir estas líneas, casi cuatro años después, me encuentro con el pelo más corto, algún kilo menos, alguna lesión más y, sobre todo, mucho a la espalda. Cómo no, debo empezar por mis directores de tesis. Luis y Manolo. La condición absolutamente necesaria para el éxito de esta tesis. Lo dije una vez hace ya algún tiempo y ahora reitero lo entonces dicho. Su calidad matemática es enorme. Agradeceros todo lo aprendido. El rigor en el escribir. Y en el pensar. Por su clarividencia vertida sobre la maraña de conceptos y teoremas. Por sus ideas. Por encontrar siempre la pregunta adecuada. Por guiar con claridad y precisión hacia la respuesta. En definitiva, por transmitir su pasión por el hacer matemáticas. Todo ello me ha permitido crecer y madurar matemáticamente, sembrando en mí la confianza y el hambre de lanzarme a nuevos desafíos. También he de agradecerles su paciente, esmerada y meticulosa lectura de la tesis. Por el tiempo invertido en la misma y sus (no pocas) correcciones. El aspecto actual de la tesis habría sido imposible sin ellos. Su lado humano es, si cabe, aún mayor. Sois dos grandes personas. Tremendamente generosos con vuestro tiempo. Genuinamente preocupados por vuestros estudiantes, velando por su bienestar y brindándoles apoyo. Vuestra flexibilidad, especialmente en mis idas y venidas a Madrid, accesibilidad y disponibilidad semanal, ya sea presencial u online, son encomiables. Vuestra labor como directores ha sido digna de ser imitada. Honestamente lo pienso. Gracias por todo. Je veux aussi profiter de ces lignes pour remercier Pascal Lefèvre et Hervé Queffélec pour m’accueillir pendant quatre mois à Lille. Faire des maths avec vous a été (et continue à être) un vrai plaisir et une expérience stimulante. J’ai vraiment profité de nos discussions, premièrement à la Salle Kampé, puis sur zoom, pour finalement se rencontrer en décembre à Sevilla. Merci de partager votre temps avec moi aussi généreusement pendant, déjà, plus d’un an. En particulier, je veux aussi remercier Hervé pour se prêter à être un des rapporteurs de la thèse, ainsi qu’à être membre du jury de la thèse. Agradicimientos I extend my gratitude to Professor Daniel Carando, for accepting being informant for my thesis, as well as member of the evaluating committee. I am also thankful to Professor Andreas Defant, Professor Pavel Gumenyuk, and Professor Pablo Sevilla for accepting to be part of the thesis committee. Let me also thank Pablo for initially accepting to be an informant, although, eventually, we did not seem to be very well-informed. I want to also thank Athanasios Kouroupis for his insight into our problem during a conference held in Thessaloniki. His remarks allowed us to significantly improve the statement of Theorem 1.5.18. Thanks also for accepting to come to Sevilla in February. He is not only a very promising young mathematician, but a truly nice person. I wish him the best. Hope to live another Carnaval together. Al personal de Secretaría del IMUS, Teresa, Carmen, Maribel, Clarines, Charo y María del Mar. Por su diligencia y su constante predisposición a ayudar, bien sea con dietas, reservas de seminarios, papeleo o invitados. Sin ellas, todo sería mucho más complicado. Debo hacer mención especial a Nacho, el logmaster. Que la tesis haya finalmente compilado se lo debo, en buena medida, a él. Gracias por ser tan generoso con tu tiempo, arreglando todos los logs y los innumerables overfulls. Final boss superado. Llega el momento de cruzar Despeñaperros y dirigirse a la Meseta. A mis queridos amigos del barrio y del colegio. Camada. Son ya unos pocos años de amistad y esta etapa que termina es otra más junto a vosotros. Todavía recuerdo aquel día de junio de 2020, reunidos, al filo de la legalidad, en una terraza vallecana y en la que os anuncié mi marcha a Sevilla. A lo largo de estos años os he echado de menos. Pese a ello, las periódicas visitas a Madrid lo han hecho más fácil, viniendo a confirmar algo sobradamente conocido: que, con vosotros, el tiempo no pasa. Sevilla es la del color especial. Pero Embajadores también. Y ese colorido se lo dais vosotros. También he tenido la oportunidad de enseñaros mi Sevilla, con el esquema tradicional de viaje de ‘fuego en la primera noche y a aguantar lo que queda, camaradas’. El pescaíto en la Plazuela acompañados de unas pocas botellas de Barbadillo son ya tradición. Un arrocito a las 5am, un asalto a la nevera de la playa, una excursión a la playa algo insomnes, un Halloween, un par de ferias y algo de turismo. Eso sí, queda un último baile. Y sois todos titulares. A mis amigos de la carrera, Manuel y Guzmán que, en nuestra rutina de jueves y litros, yoga y parques, kiosko y cafetería, teoremas y alguna clase, nos arrimamos a Óscar, Antonio, Samuel y Vicen, conformando este pintoresco, diverso y entrañable grupo. Por todos estos guateques en San Modesto, Aranda, Murcia, Sevilla, Roma, Caraquiz, Málaga o Albania. Y por los que, pese a la diáspora geográfica, seguiremos celebrando. Y porque éstos sigan marcados por la habitual intensidad en el goce, en la conversación y en la música. A Edu que, desde chiquitito con sus trotes, primero por Madrid y, luego por el mundo, nos lleva a descubrirlo junto a él. Querido amigo, nos vemos pronto en la próxima aventura. A las Sandras: Sansi y Sanno. Por hacerme un hueco siempre que les escribía para acercarse al barrio a darme un abrazo. Por encontrar ratos, por cortos que fueran y por escaso que fuera el aviso, para ponerse al día. A Guille, porque, por muy pequeña que sea la intersección de nuestras agendas, siempre acabamos encontrándola. Y siempre con buena comida y bebida de por medio con la que acompañar las novedades vitales. A Dani, por ser tan fiel, genuino y risueño. Por ser un gran amigo y un mejor padre. A mis queridos amigos en la península Itálica, Jonny y Troy. Jonny, respirando a través de la comedia. Fuente de discusión perpetua y carcajada constante. Un saludable contrapeso. Un cómplice y un confidente. Por enseñarme su hermosa isla y a su gente. Troy y su gran corazón. Por su atenta escucha. Por dar equilibrio y aguantar las machangadas de Jonny y de un servidor. Por su infinita fidelidad, amabilidad y hospitalidad. Y, por qué no, por su limoncello de todas las frutas imaginables. Nos vemos pronto. Acabaremos convergiendo en el espacio y en el tiempo. De vuelta a la ciudad del Guadalquivir, mis primeros pasos en Sevilla fueron en ese piso de (Juan) Sebastián Elcano. Esas paredes fueron testigos del inicio de una duraderas y hermosas amistades. Como las de mis, en aquel entonces, compañeros de piso, ahora queridos amigos, Roberto y Pablo. Gracias por el cariñoso y exquisito trato que, desde el primer día, me habéis dado a mí y a los míos. A partir de ahí, las amistades se sucedieron de manera natural. Mis muy queridas Mari y Carmen, fuente constante de alegría, cariño y diversión. Güili, con su ocurrencia, su jovialidad y su contagioso y fino sentido del humor. Dani, compañero de excursiones en el inclemente invierno noruego y camarada de remo, ya fuera bajo un sol inclemente o un frío húmedo. Si mi llegada e instalación en Sevilla fueron tan fáciles fue, sin duda, gracias a todos ellos. Ese piso contiene una infinidad de bonitos recuerdos, de los cuales sois protagonistas. Juntos conformamos una cuadrilla de guateques para el recuerdo. Con sus disfraces, sus cuencos, sus vinos, sus bailes, sus picnics, sus quesos y sus papas. Ellos me abrieron las puertas de un tejido social envidiable, que me acogió como uno más. Quiero agradecer a las familias Arjona, Calle y Mercé Fernández. Por haberme tratado con tan inmenso cariño. Por haberme hecho sentir siempre como uno más. A mí y a los míos. Por su inmensa bondad. Los momentos vividos junto a vosotros los guardo con un cariño especial. Gracias. No puedo seguir sin antes hablar de la tropa del IMUS. Y es que no son pocas las personas que he conocido a lo largo la tesis. El haberlo hecho me llena de felicidad. Desde mi llegada al silencioso despacho 10 y mirada fija en la pared, hasta el bullicioso trasiego del despacho 8. Las comidas en la terraza, los cafés de máquina, los desayunos y los cafés en el Brasil resultan verdaderamente terapéuticos. También ha habido tiempo para viajar, aunque no sea a una casa rural, hacer excursiones, comidas y comidas que se convierten en cenas. Los Metrolíos fueron la principal fuente de ocio en plenas restricciones. También se hacía tesis entre Jamesons. Por los supervivientes refugiados en Virgen del Valle entre pizzas y cruzcampos. Las azoteas, los amigos invisibles y las risas Agradicimientos grabando las escenas de los que iban defendiendo. Por las recogidas, con o sin propina. Por las palas. Por los hall of famers, por la next gen y por la old gen, cada día más old, pero lejos de jubilarse. Todo ello conforma un maravilloso mosaico de recuerdos imborrables. También acordarme de los que se encuentran allende los Pirineos. Vuestras visitas me alegran el corazón. Tampoco quiero olvidarme de mi profesor, sensei y amigo José Luis. Por mostrarme con su hermoso y elegante judo que queda mucho por aprender. Por su preocupación genuina en mis lesiones. Por ponerme siempre en las mejores manos. Mil gracias por todo. Y a Nacho. Con quien más horas de tatami he compartido en Sevilla. Por llevarme y traerme. Por su disponibilidad y su extraordinaria generosidad. Por enseñarme el BJJ. Por las siempre bonitas palabras hacia mi persona. Por las comidas y las cervezas post entreno. En definitiva, por ser así. A quien, pese a lo que implicaba la mudanza a Sevilla, no dejó de apoyarme en cada paso. A ella. A Isabel. Muchos han sido los blablacar, los aves, los vuelos, las videollamadas, los mensajes. Los jueves-domingo. Las cábalas calendario en mano. Los planes y las vacaciones. Las indeseadas despedidas y los sentidos reencuentros. Por su presencia en la ausencia. Por alegrarse en mis éxitos. Por su persistente presencia en los baches. Por todo lo vivido en estos años. Por sus frecuentes visitas a Sevilla. Por haber abrazado mi mundo aquí, y a los míos como suyos. Y hacer, por su ser tan especial, que acabaran siendo los suyos también. Por su saber escuchar. Por su saber cuidar y su saber querer. Por el ayer y por el hoy. Por el mañana. También es inmensa mi gratitud hacia su familia. Maribel, Antonio y Pablo. Por su sincera y genuina preocupación por mí y los míos. Por su exquisito trato hacia mí en todo momento y ocasión a lo largo de todos estos años. A mi familia. Qué deciros. Mi agradecimiento trasciende, con mucho, estos años en Sevilla. Os debo lo que soy. Y lo que tengo. Mamá y papá, me lo habéis dado todo. Y lo seguís haciendo. Vuestra bondad, vuestro afecto, vuestro amor y vuestro apoyo. Acompañándome en mis excentricidades, ocurrencias y planes. Javi, mi querido hermano y compañero de batallas, con quien todo es tan fácil. Tan cariñoso y adorable como grande. Subir a Madrid y estar en casa con vosotros siempre resulta sanador. Donde siempre. Viendo una película juntos, por mucho que cueste elegirla. Compartiendo una comida. O dando un paseo. Teneros de visita en Sevilla, por accidentadas que éstas fueran, y ver cómo la disfrutabais tanto como yo me ha hecho muy feliz. Va siendo hora de detenerse para entrar en materia. Y conviene, pues, como dice esa canción de Siniestro Total, ‘el llanto empieza a brotar’. .. Abstract The two main objects studied in this thesis are some operators acting on Banach spaces of Dirichlet series and continuous semigroups of analytic functions. The main part of thesis is mostly oriented towards the study of the first ones, although the first chapter is entirely devoted to continuous semigroups. Nevertheless, at several points along the thesis, both notions are often mixed together. Regarding the continuous semigroups { Φ t}t≥0 of analytic functions, these objects are considered in the classical setting of the unit disc D and in the right half-plane C+ . We characterise the continuous semigroups in C+ with Denjoy-Wolff point ∞ converging uniformly to the identity in the whole right half-plane as t→ 0 + . Concerning the continuous semigroups in D , we provide a quantitative version of the well-known fact that these semigroups converge to the identity uniformly in the whole unit disc. We prove that the rate of convergence is always O ( √t ), as t→ 0 + , and this order of convergence is sharp. In the part of the thesis devoted to Banach spaces of Dirichlet series several problems are considered. We begin by characterising the strongly continuous semigroups of composition operators in the Hardy spaces of Dirichlet series Hp . This is done in terms of the continuous semigroups in the so-called GordonHedenmalm class G . This class consists on the analytic functions Φ : C+→C+ giving rise to bounded composition operators on H2 . The existence of a rich variety of such semigroups is ensured thanks to the description of the infinitesimal generators of such continuous semigroups. Namely, these infinitesimal generators are those Dirichlet series sending the right half-plane into its closure. Then, we move on to the algebra of Dirichlet series, this is, the bounded Dirichlet series in C+ which are uniformly continuous there. We characterise the bounded composition operators CΦ acting on this algebra. We also show, for CΦ , the equivalence between compactness and weak compactness and provide several characterisations of this property. A description of the strongly continuous semigroups of composition operators in the algebra is also given. We conclude with the consideration of a third class of Banach spaces of Dirichlet series: a family of Bergman type spaces. Two main problems are considered in this context. First, the estimate of the norm of the evaluation functionals for a certain collection of these Bergman spaces. Second, we carry out a detailed study of the Volterra operator Tg acting on these Bergman type spaces of Dirichlet series. Introduction operators and Volterra operators. This introduction is devoted to present the main results obtained in the thesis. However, the order of appearance of these results does not correspond to the expository order of the thesis. We have preferred to follow the chronological order in which the different problems were considered and the corresponding results were obtained. We still need to present some of the objects which we have studied in our work. As it was said before, the appearance of Dirichlet series in the mathematical scene dates back to the 1800’s. During the XIX-th century, Dirichlet series attracted some attention, although its study was mostly focused in Riemann’s zeta function and applications in Number Theory. In order to find new remarkable contributions on general Dirichlet series theory, we have to move forward to 1913 and H. Bohr’s work. Bohr devoted quite some time and effort, see for instance [ 28 ], [ 30 ], or [ 29 ], to study Dirichlet series and some related problems concerning the relationship between the convergence abscissae. In fact, among many valuable contributions, probably one of his most influential ones was the now known as Bohr’s point of view. Essentially, he observed how, by the Fundamental Theorem of Arithmetic, the function defined by a Dirichlet series could be formally seen as depending on infinitely many variables. As we will see, this a priori formal observation ended up having deep and far-reaching consequences. In some unpublished notes later appeared in [ 41 ], Beurling had sketched the potential of Dirichlet series and Bohr’s point of view in order to tackle a long-standing problem in Harmonic Analysis. However, its solution had to wait until 1997, when Hedenmalm, Linqvist, and Seip were able to solve it in their seminal paper [ 64 ]. One of the most interesting points of their answer was, precisely, all the theory developed by the authors. Their work meant the birth of the theory of Banach spaces of Dirichlet series. Indeed, they introduced the first two Banach spaces of Dirichlet series: the Hilbert space H2 and the space of bounded Dirichlet series in C+ , H∞ . One key ingredient in [ 64 ] was the realisation that Bohr’s point of view was far more than just a formal correspondence. The different results proven in [ 64 ] showed that these spaces exhibited some exotic properties and presented an extremely rich structure, manifesting the intrinsic interest of their study. In fact, in this seminal work, a first dualism, whose importance has been proven to be essential, appeared. Indeed, the space H∞ is a Banach space on the half-plane C+ := {s∈C : Res > 0 } , whereas the space H2 consists of analytic functions defined in the half-plane C1/2 := {s∈C : Res > 1 / 2 } . The abscissa Re ( s )=0 and the abscissa Re ( s )=1 / 2, as we will see throughout the exposition, play a capital role in the theory of Hp-spaces. One of the topics studied when dealing with Banach spaces of analytic functions is the behaviour (boundedness, compactness, weak compactness, etc.) of certain operators acting on the spaces. A well-known and deeply studied operator is the composition operator CΦ . Given a Banach space X of analytic functions on a domain Ωin the complex plane C and a holomorphic function Φ : Ω → Ω, the composition operator CΦ of symbol Φis defined by CΦf = f◦ Φ, f∈X. ii If we take X = H2 , then Ω = C1/2 . Hence, in such case, for a composition operator CΦ to be well-defined in H2 , we have to consider an analytic symbol Φ : C1/2→C1/2 . The study of the characterisation of the boundedness of the composition operator CΦ in H2 led to another remarkable paper [ 61 ] from 1999 signed by Gordon and Hedenmalm, where such characterisation was established. The description was given in terms of the now known as Gordon-Hedenmalm class, denoted by G . This class consists on those analytic functions Φ : C+→C+ such that they can be represented as Φ(s) = cΦs+φ(s),(1) for some cΦ∈N∪{ 0 } and some Dirichlet series φ . Moreover, if cΦ = 0, we also require that Φ( C+ ) ⊂C1/2 . This last restriction arises from the fact that functions in H2 are defined in the half-plane C1/2 ; so it has to be imposed to guarantee the correct definition of CΦ . Then, the operator CΦ with symbol Φ : C1/2→C1/2 defines a bounded composition operator on H2 if and only if Φhas a holomorphic extension to C+belonging to the class G. Once more, another even more surprising interplay between the half-planes C+ and C1/2 emerges. Indeed, for an analytic symbol Φ : C1/2→C1/2 to induce a bounded composition operator CΦ on H2 , despite being a space defined on C1/2 ,Φneeds to have an analytic extension to the whole right half-plane satisfying certain properties. The boundedness of the composition operator CΦ on H∞ was addressed by Bayart in 2002, see [ 16 ]. There, he showed that CΦ is bounded on H∞ if and only if there exist cΦ∈N∪{ 0 } and φ∈ D such that Φcan be represented as Φ( s ) = cΦs + φ ( s ). We denote this class of symbols by G∞ . In this class there are no more mapping restrictions when cΦ = 0, since the functions in H∞ are defined in the whole right half-plane C+ . In fact, this implies that the class G is strictly contained in the class G∞ . In particular, this means that, in spite of H2 being a far larger space than H∞ , there exist analytic symbols Φ : C+→C+ defining bounded composition operators CΦ on H∞ , but failing to be bounded when considered acting on the space H2. The spaces H2 and H∞ were little after embedded in a larger family of Banach spaces of Dirichlet series. Indeed, in 2002, the picture was completed by Bayart [ 15 ] with the introduction of the Hp spaces of Dirichlet series for the range p∈ [1 ,∞ ). The definition of these spaces was based on Bohr’s observation and on the realisation, already done by Hedenmalm et al. for the hilbertian case, that Bohr’s correspondence was more than formal. These key ingredients worked together for defining a whole new family of spaces of analytic functions in C1/2 . Essentially, by lifting the Dirichlet polynomials to the infinite polydisc D∞ and to the infinite polytorus T∞ , Bayart was able to define an Lp ( T∞ )-norm for these ‘lifted’ polynomials of infinitely many variables. Then, the spaces Hp are defined as the completion of the Dirichlet polynomials under this norm. The most delicate part consists on verifying that the elements of the completion can be realised as Dirichlet series convergent in C1/2 . These novel spaces posed new intriguing and stimulating questions, as well as demanded new techniques and tools differing from the ones typically used in the classical setting of the unit disc D . Ever since the appearance of this new family of Banach spaces, the interest for these spaces, in particular, and for the theory of Banach spaces of Dirichlet series, in general, has done nothing but increase. iii Introduction In the same work, Bayart also studied the validity of the Gordon-Hedenmalm class in order to describe the bounded composition operators on Hp ,1 ≤p < ∞ . It happened that the description of these symbols is significantly more delicate than for the case p = 2 and it is far from being completely solved. Bayart showed that, for the case cΦ∈N , the characterisation given by Gordon and Hedenmalm for H2 still works for the range 1 ≤p < ∞ . Furthermore, the operator CΦ acts contractively on Hp . However, the problem becomes more delicate when cΦ = 0. In fact, Bayart and Brevig have shown [ 19 ] that giving this characterisation is equivalent to solving the local embedding problem (see [ 75 , pp. 177-178]); a key ingredient for further development of the theory of the spaces Hp . Bayart proved that the membership in the Gordon-Hedenmalm class was necessary for boundedness in Hp . If p∈ 2 N , the condition is known to be sufficient. However, for p > 2and p∈ 2 N , Gordon and Hedenmalm’s condition is not known to be sufficient. Not only this but, as pointed out first by Bayart and Brevig [ 19 ], and also as a consequence of a later deep result due to Harper, for the range 1 ≤p < 2the membership in the class G is no longer sufficient. A second object of capital importance in this thesis is that of continuous semigroups of analytic functions. Let us recall that given a region Ω ⊂C , a family { Φ t}t≥0 ( { Φ t} for short) of holomorphic self-maps of Ωis a semigroup if Φ 0 ( z ) = z , for every z∈ Ω, and Φ t◦ Φ u = Φ t+u , for every u , t≥ 0. Moreover, if Φt(z)→z , as tgoes to 0+, uniformly on compact subsets of Ω, we say that { Φ t} is a continuous semigroup in Ω. There exists a sort of analogue version of the continuous semigroups for the case of families of bounded operators. A family {Tt}t≥0 ( {Tt} for shorter) of bounded operators from a Banach space X into itself is said to be a semigroup if T0 = Id , where Id denotes the identity map on X , and if for every t, u ≥ 0, Tt◦Tu = Tt+u . If, in addition, it satisfies that limt→0+Ttf = f for all f∈X, we say that it is a strongly continuous semigroup. The composition operators yield a bridge between the semigroups of analytic functions and the semigroups of composition operators. Clearly, if we consider a semigroup { Φ t} of analytic selfmaps of Ω, then the family {Tt} , with Tt = CΦt , is a semigroup of operators. The converse is also true. When dealing with Dirichlet series, this (algebraic) correspondence between semigroups of symbols and semigroups of operators still holds, although one of the implications is less immediate. Berkson and Porta, in their fundamental paper from 1978, showed the one-to-one correspondence existing between continuous semigroups of symbols in D and strongly continuous semigroups of composition operators in the Hardy spaces Hp ( D ),1 ≤p < ∞ . In this case, X = Hp and Ω = D . The first problem of this thesis [ 22 ] was to consider X = H2 and Ω = C1/2 and seek for an analogue of Berkson and Porta’s result in this new setting. The algebraic correspondence between semigroups of symbols and semigroups of operators holds, as pointed out above. Actually, the fact that a semigroup { Φ t} of analytic symbols in the class G is continuous is equivalent to the strong continuity in H2 of the family {Tt} of bounded composition operators, Tt = CΦt . Not only this, but, surprisingly, the continuity of the semigroup { Φ t} is actually equivalent to the convergence of the semigroup to the identity uniformly on half-planes iv Cε := {s∈C : Res>ε} , for all ε > 0(see Theorem 3.2.6 for a stronger result). At that moment, it was not clear for us if such uniform convergence holds in C+. We shall come back to this problem later in the introduction. This Berkson-Porta type theorem for the hilbertian case p = 2 passes to the range 1 ≤p < ∞ . The argument is essentially the same, being a key ingredient that the elements of a continuous semigroup in the class Gare of the form Φt(s) = s+φt(s), s ∈C+. That is to say, the characteristics cΦt of the elements Φ t of a continuous semigroup { Φ t} in the class G are, for all t > 0, equal to 1(see Proposition 3.1.5). This fact is crucial since, as the characterisation of the boundedness of CΦ on Hp remains the same for cΦ∈N and, in addition, the composition operator is a contraction on Hp ,1 ≤p < ∞ , we have a uniform control of the operator norm for every t > 0. This allows us to pass to the whole range of finite pin Theorem 3.2.8. The case p = + ∞ , as in the case of the unit disc, is an example of the failure of this correspondence between continuous semigroups of symbols and strongly continuous semigroups of composition operators. In fact, in Theorem 3.4.1 it is proven that the only strongly continuous semigroup of composition operators in H∞is the trivial one, namely, Tt=Id, for all t > 0. In Berkson and Porta’s paper another remarkable result was proven. They showed the intrinsic relationship between the continuous semigroups and the theory of differential equations. Let { Φ t} be a continuous semigroup in Ω. Given z∈ Ω, the map t7→ Φ t ( z )is known as the trajectory of the semigroup starting at z . Berkson and Porta showed the existence of a unique holomorphic function H : Ω →C such that the trajectories are the solution to the initial value problem ∂Φt(z) ∂t =H(Φt(s)) and Φ0(z) = z∈Ω.(2) In fact, we have H(z) = lim t→0+ Φt(z)−z t,for all z∈Ω,(3) where the convergence is locally uniform in Ω. The holomorphic function H is called the infinitesimal generator of the semigroup { Φ t} . Berkson and Porta provide in their work several characterisations of the infinitesimal generators of continuous semigroups in the unit disc and in the right half-plane. Hence, according to the previous result, given a holomorphic function H on a domain Ωwith the adequate properties, we can recover the associated semigroup by solving the Cauchy problem (2). Another key notion in the theory of continuous semigroups of analytic functions is the Denjoy-Wolff point of a continuous semigroup { Φ t} . Let Ω be either D or C+ and assume that { Φ t} does not consist on automorphisms having a fixed point in Ω. The Denjoy-Wolff Theorem asserts that for a continuous semigroup in Ω, there exists a unique point τ∈Ω∪ {∞} such that limt→+∞ Φ t ( s ) = τ , for all s∈ Ω. In the case Ω = C+ , Berkson and Porta actually provided a description of those functions H : C+→C v Introduction that are infinitesimal generators of continuous semigroups in C+ with DenjoyWolff point ∞ . These functions are all the holomorphic functions satisfying H(C+)⊂C+\{0}. At this point, a natural question emerges. So far, we have seen that the continuous semigroups in G characterise the strongly continuous semigroups in Hp ,1 ≤p < ∞ . A clear example of a continuous semigroup { Φ t} satisfying this theorem is, fixed a with Re ( a ) ≥ 0, the semigroup of translations Φ t ( s ) = s + at , s∈C+. However, are there other continuous semigroups in the class G? It is not clear by simple inspection that one can provide non-trivial explicit examples of such semigroups. Nonetheless, a detour through Berkson and Porta’s theory proved to be far more fruitful. We know that the continuous semigroups in the class G fall in the larger class of continuous semigroups in C+ whose Denjoy-Wolff point is ∞ (see Proposition 3.1.10). If we consider a holomorphic function H : C+→C+\{ 0 } , we know that the solution to the Cauchy Problem (2) is given by the trajectories of a continuous semigroup in C+ . Now, if, in addition, H can be represented by a convergent Dirichlet series, can we guarantee that the continuous semigroup { Φ t} consists of elements in the class G ? This is actually the case and, to prove this statement, it is necessary to use an adapted Cauchy-Picard type argument. In fact, this sufficient condition is also necessary (see Theorem 3.3.2). This is, given a continuous semigroup { Φ t} in the class G with infinitesimal generator H : C+→C+\{ 0 } , we have that H can be represented as a convergent Dirichlet series. As a consequence of Theorem 3.3.2, there is a wide range of examples of continuous semigroups in G. Another Banach space of Dirichlet series considered in this thesis is the algebra of Dirichlet series A ( C+ ), less studied in the literature. This space is the subspace of H∞ consisting on those functions which are uniformly continuous in the right half-plane C+ . The introduction of this space dates back to 2017 and it is due to Aron, Bayart, Gauthier, Maestre, and Nestoridis, see [9]. The first remarkable difference with respect to the disc algebra i.e. the holomorphic functions in D which are uniformly continuous, is that, whereas the disc algebra coincides with the holomorphic functions in D which have a continuous extension to D , the algebra A ( C+ )differs from the subalgebra of H∞ of functions having a continuous extension to C+(see Proposition 4.1.7). The first problem considered in the algebra is the characterisation of the bounded composition operators CΦ acting on A ( C+ ). Such description is provided in terms of a class, denoted by GA , which is actually the subclass of G∞of symbols Φ : C+→C+uniformly continuous in the sets AM={s∈C:Re(Φ(s)) < M},for all M > 0. Then, a composition operator CΦ is bounded on A ( C+ )if, and only if, the symbol Φbelongs to the class GA (see Theorem 4.2.2). This class is in fact strictly contained in G∞ and contains strictly the class of symbols belonging to G∞ which are uniformly continuous in C+ (see Example 4.2.4 and Example 4.2.5, respectively). Afterwards, the compactness and weak compactness of these operators on A ( C+ )is addressed. Both notions happen to be equivalent. This is, a vi composition operator CΦ is compact on A ( C+ )if and only if it is weakly compact. For this to occur, the symbol Φmust map the right half-plane into a strictly smaller half-plane. Not only this, but the compactness is also equivalent to not fixing a copy of the sequence space c0 (see Theorem 4.3.5). This alternative description in terms of not fixing copies is achieved using some ideas involving interpolating sequences in the disc algebra. In fact, these ideas can be also used to deduce an analogue characterisation of the compactness and weak-compactness in H∞ in terms of not fixing a copy of the non-reflexive sequence space ℓ∞ (see Theorem 4.3.6). This result completes the picture of some previously known results from Bayart [17] and Lefèvre [70]. Once the boundedness of the operator CΦ on A ( C+ )is settled, we consider once more the Berkson-Porta problem in this new setting: is the continuity of a semigroup in the class GAequivalent to the strong continuity in A(C+)of the semigroup {Tt} of composition operators Tt = CΦt ? It is not very difficult to see that the answer to this question is affirmative if and only if we prove that every continuous semigroup in GAconverges to the identity uniformly in C+. Question 1. Given a continuous semigroup in G∞ , is it true that the converges to the identity uniformly in C+ ? If not, is it true, at least, for continuous semigroups in GA? We soon realised that the techniques needed to face this question did not involve Dirichlet series theory. Hence, we considered the problem in the more general setting of continuous semigroups of analytic functions in C+. Question 2. Given a continuous semigroup in C+ with Denjoy-Wolff point ∞ , does it converge to the identity uniformly in C+? Berkson and Porta’s description of the infinitesimal generators of such semigroups allows to describe many properties of the semigroup in terms of its infinitesimal generator. Because of this, it is no surprise that the answer to the previous questions is given precisely in terms of such objects. The turning point to tackle these problems was the use of the powerful tool of harmonic measure together with a theorem due to Lavrentiev. Carrying out a delicate construction, we are able to actually describe the continuous semigroups { Φ t} in C+ whose Denjoy-Wolff point is ∞ converging uniformly to the identity in C+ . Such convergence happens to be equivalent to the uniform convergence in Cε , for some ε > 0– and then for any ε > 0– (see Theorem 1.5.18). Hence, the answer to Question 1 is affirmative, whereas the answer to Question 2 is negative, in general. In fact, in Theorem 1.5.18, we also provide a characterisation of the semigroups for which the answer to Question 2 is affirmative in terms of the infinitesimal generator of the semigroup. The ideas and the techniques involved in the proof of these results can be used to derive a relevant consequence for the continuous semigroups in the unit disc D . Indeed, in 2005, Contreras and Díaz-Madrigal showed that every continuous semigroup of analytic functions in D belonging to the disc algebra converges uniformly to the identity in D . In 2014, Gumenyuk [ 62 ], proved that, in fact, this statement holds for every continuous semigroup in the unit disc D . In Theorem 1.6.2, we are able to provide a quantitative version of Gumenyuk’s theorem, proving that the rate of convergence is O ( √t ). Moreover, vii Introduction this rate is optimal, see Proposition 1.6.4, in the sense that there exist continuous semigroups {Φt}in Dsuch that lim inf t→0+ supz∈D|Φt(z)−z| √t>0. Coming back to semigroups of composition operators on Hp and on A ( C+ ), our result for continuous semigroups of analytic functions allows us to complete the proof of Theorem 3.2.6. Moreover, we can also use it to prove that a semigroup of composition operators in A ( C+ )is strongly continuous if and only if the continuous semigroup of symbols is in GA(see Theorem 4.4.3). One of the most natural questions arising when studying Banach spaces of Dirichlet series is that of computing or, at least, estimating the norm of the evaluation functionals. We recall that given a Banach space Xof analytic functions in a domain Ωin the complex plane C and a point s∈ Ω, the evaluation functional at the point s , denoted by δs , is defined as δs ( f ) = f ( s ), f∈X. The next problem studied in the thesis is that of providing a norm estimate of the functional δs taking as Banach space X = Ap α ,1 ≤p < ∞ , α > − 1. These spaces were originally introduced in 2015, by Bailleul and Lefèvre [ 14 ], generalising a first incursion into defining Bergman spaces analogues for Dirichlet series, done in 2004 by McCarthy [71], but just for the hilbertian case. The spaces Ap µ , where µ is a probability measure on (0 ,∞ )such that 0 ∈supp ( µ ), are defined as the completion of Dirichlet polynomials under the Ap µ-norm, defined for a Dirichlet polynomial fas ∥f∥Ap µ=Z∞ 0∥fσ∥p Hpdµ(σ)1 p , where fσ ( s ) = f ( s + σ ). The spaces Ap α correspond to the following choice of the probability measure µ: dµα(σ) = 2α+1 Γ(α+ 1)σαe−2σdσ, α > −1. In [ 15 ], Bayart was the first to address the problem of computing or estimating the norm of δs in the Hp -spaces, p∈ [1 ,∞ ). Indeed, for s∈C1/2 , Bayart showed that ∥δs∥p (Hp)∗=ζ(2Re(s)). For the case of the spaces Ap α , p∈ [1 ,∞ )and s∈C1 2 , the following upper estimate was proven in Bailleul and Lefèvre’s work: ∥δs∥(Ap α)∗≲Re(s) Re(s)−1/2α+2 p, α > −1. For α∈ ( − 1 , 0), Bailleul and Lefèvre also showed that the lower estimate of the norm by a term equivalent to the right hand side still holds. However, for α≥0, the only known (until now) lower estimate was the following: ∥δs∥(Ap α)∗≳1 (Re(s)−1/2)α+2 p(1 + |log(2 Re(s)−1)|)1 p· viii Let us point out that the consideration of the point evaluation δs in the halfplane C1/2 is justified by the fact that, as in the case of the Hardy spaces Hp , the spaces Ap µare Banach space of analytic functions in this half-plane. The main result of Chapter 5extends the norm estimate of the functionals δs from the range α∈ ( − 1 , 0), to the range α > − 1(see Theorem 5.4.2). More precisely, for all α > −1: ∥δs∥(Ap α)∗≈Re(s) Re(s)−1/2α+2 p, s ∈C1 2.(4) This also provides an affirmative answer to Question 2 from [55]. The proof of estimate (4) involves the Riemann-Liouville operator It . This operator acts on Dirichlet series as a fractional integration operator. Having this, one key point is that the operator It allows to write the Ap α -norm of a Dirichlet series f in terms of the norm of Itf in some space Ap β . This property will play a crucial role in passing the norm estimate from the range α∈ ( − 1 , 0) to the whole range α > −1. For a convergent Dirichlet series g ( s ) = Pn≥1ann−s , the associated Volterra operator of symbol g , denoted by Tg , acting on a Dirichlet series f(s) = Pn≥1bnn−sis defined by the integral f7→ −Z+∞ s f(w)g′(w)dw. A natural problem is the study of the boundedness of Tg acting on a certain Banach space. To the best of the author’s knowledge, the first systematic study of the boundedness of the operator Tg acting on Banach spaces of Dirichlet series was carried out in [ 39 ] by Brevig, Perfekt, and Seip. There, a sufficient condition for boundedness on Hp was given. Namely, the membership of g in BMOA ( C+ ). It was also established that whenever Tg is bounded on Hp and, surprisingly, p∈Q+ , then g is a Dirichlet series belonging to the space BMOA ( C1/2 ). Moreover, it was shown that the sufficient condition fails to be necessary and viceversa. After this first work on the topic, several papers studying the operator Tg acting on other Banach spaces of Dirichlet series have been written. In [ 31 ], Bommier-Hato gave a sufficient and a necessary condition for the boundedness of Tg on the Hilbert version of a different family of weighted Bergman spaces of Dirichlet series, namely, the spaces Hp w . Bommier-Hato showed that the membership of g to the Bloch space Bloch ( C+ )is sufficient to guarantee the boundedness of Tg on H2 w , as long as its Bohr lift depends on finitely many variables. In the same work, she also gave a necessary condition for boundedness on H2 w in terms of the space Bloch ( C1/2 )of Dirichlet series. Regarding the sufficiency, very recently, Chen and Wang in [ 42 ] were able to extend the range of p to 1 ≤p < ∞ , as well as to remove the previous restriction on the Bohr lift of the symbol. Nevertheless, they give more restrictive conditions on g . Chen and Wang show that the membership of the symbol g to the smaller space BMOA ( C+ )is sufficient for the boundedness of Tg on Hp w . On the other hand, for the case of 1-homogeneous symbols g, this is, when gis of the form g(s) = X pprime app−s, ix Introduction they prove that Tgis bounded on Hp wif and only if g∈ H2 w. Regarding the spaces Ap µ , the first study of the operator Tg in this setting appeared in [ 55 ]. More precisely, Fu, Guo, and Yan considered a specific family of measures να , α > 1. Let us point out that these measures are not probability measures, not even finite measures. Instead, they are σ -finite, but they still satisfy that 0 ∈supp ( να ). Despite this, the authors are still able to define the spaces Ap να in the same fashion as in the case of the spaces Ap µ . In fact, when removing the constants, the family of measures µα and να , which differ on an exponential term, generate the same spaces Ap µ (see Lemma 5.2.11 and the comment done right after). Not only this, but for each α , the space Ap να is the hyperplane in Ap µα−2 consisting on those functions with vanishing first coefficient. This is particularly relevant when studying the boundedness of the Volterra operator Tg . Regarding the study of the operator Tg acting on Ap να , Fu et al. give a sufficient condition for the boundedness of Tg on Ap να ,1 ≤p < ∞ , namely, the membership of the symbol g to the space Bloch ( C+ ). They also prove that whenever Tg acts boundedly on the Hilbert space A2 να , the symbol g is a Dirichlet series in the space Bloch(C1/2). We shall give a sufficient condition for the boundedness of Tg on Ap µ depending on the measure µ . More precisely, it will be sufficient for the symbol to belong to the Bloch-type space Blochµ ( C+ )(see Theorem 6.4.1). In particular, being Ap να a complemented subspace in Ap µα , the sufficient condition given in [ 55 ] is obtained as an immediate consequence of this result. In Theorem 6.4.9, we also show that the membership to Bloch ( C1/2 )is a necessary condition for the boundedness of Tg acting on the special case spaces Ap α , where α > − 1 and p∈ [1 ,∞ ). This improves the already known results from [ 55 ] by extending both the range of p and α . The proof of the necessity is based on the estimate of the norm of the pointwise evaluation functionals in Ap α from Chapter 5(see Theorem 5.4.2). Organisation of the thesis. The thesis comprises six chapters. Chapter 1consists on two parts. A first introductory exposition devoted to present all the background needed from the theory of continuous semigroups of analytic functions in the unit disc D and the right half-plane C+ . In the second part of the chapter, we present all the new results concerning the convergence to the identity of continuous semigroups in C+ and the quantitative version of Gumenyuk’s Theorem. The proof of this last result is first carried out for the elliptic case and then for the non-elliptic. Chapter 2is essentially introductory. A first general review of the most basic properties of Dirichlet series like abcissae of convergence, vertical limits or horizontal translations is presented. Then, the construction of the Hp spaces and the Ap µ spaces is recalled. A section is devoted to recapitulate the so far known results regarding the boundedness of composition operators on the spaces Hp . The unique original part of this chapter is Section 2.4, where the closeness of the classes Gand G∞under local uniform convergence in C+is studied. Chapter 3contains the first problem of the thesis presented above. Namely, a characterisation of the strongly continuous semigroups of composition operators on Hp ,1 ≤p < ∞ . Also, the characterisation of the infinitesimal generator and the Koenigs function of continuous semigroups in the class Gis proven. x In Chapter 4, the ambient space is the algebra of Dirichlet series A ( C+ ). There, all the results concerning the boundedness, compactness, and weak compactness of the operator CΦ on A ( C+ )are presented. The second part of the chapter contains the description of the strongly continuous semigroups of composition operators on A ( C+ ), as well as some interesting related questions. Chapter 5is devoted to present the techniques and results needed in order to prove the norm estimate of the evaluation functionals in the spaces Ap α . We will also see how the Riemann-Liouville operator It is used to characterise the injection of Hpin Aq α. Chapter 6is dedicated to the study of the Volterra operator Tg acting on the spaces Ap µ . We begin the chapter by presenting a new family of spaces of Dirichlet series, namely, the spaces Blochµ ( Cθ ), θ≥ 0. After proving some key properties of these Bloch-type spaces, we will establish a Littlewood-Paley formula for the Ap µ -norm. This being done, the rest of the chapter is devoted to the proof of the necessary and the sufficient conditions for the boundedness of Tg on Ap µ . The proof of the sufficient condition is given by proving a Carleson measure type condition. The proof of the necessity relies both on estimate (4) and on an upper estimate of the norm of the evaluation of the derivatives in the spaces Hp. The original results presented in the thesis have been extracted from the following research works: • [ 48 ]: M.D. Contreras, C. Gómez-Cabello, and L. Rodríguez-Piazza, Semigroups of composition operators on Hardy spaces of Dirichlet series, J. Func. Anal. 285 (2023), paper no. 110089, 1–36. • [ 47 ]: M.D. Contreras, C. Gómez-Cabello, and L. Rodríguez-Piazza, On the uniform convergence of continuous semigroups, preprint, https: //arxiv.org/pdf/2304.12759.pdf, (2023). • [ 49 ]: M.D. Contreras, C. Gómez-Cabello, and L. Rodríguez-Piazza, Composition operators on the algebra of Dirichlet series, preprint, https: //arxiv.org/pdf/2311.18790.pdf, (2023). • [ 59 ]: C. Gómez-Cabello, P. Lefèvre, and H. Queffélec, Integration type operators and point evaluation on weighted Bergman spaces of Dirichlet series, preprint, https://arxiv.org/pdf/2402.12523.pdf, (2024). • [ 60 ]: C. Gómez-Cabello, P. Lefèvre, and H. Queffélec, Volterra operator acting on Bergman spaces of Dirichlet series, preprint, https://arxiv.org/ pdf/2402.12524.pdf, (2024). The research work [ 47 ] contains the results exposed in Chapter 1. In [ 49 ], we prove the original section contained in Chapter 2and the results appearing in Chapter 4. In [ 48 ], we prove the results from Chapter 3. The preprint [ 59 ] includes the results presented in Chapter 5. Eventually, [ 60 ] contains the results treated in Chapter 6, as well as further questions such as compactness or membership to Schatten classes which do not appear in the thesis. xi 1. Continuous semigroups of analytic functions generator H and the Julia-Wolff-Carathédory Theorem ([ 35 , Theorem 1.7.3]). Indeed, applying this result, we have that, Re(Φt(z)) ≥αRe(z), z ∈C+, for some α≥1. Therefore, Re(H(z)) = lim t→0+ Re(Φt(z)) −Re(z) t≥0. 1.4 The Koenigs function In this section, we introduce the Koenigs function of a continuous semigroup. This object, together with the infinitesimal generator of the semigroup, is a key notion in the study of the continuous semigroups and some of its most essential properties. The Koenigs function linearises simultaneously all the iterates of the continuous semigroup. Theorem 1.4.1. ([ 35 , Theorem 9.3.5]) Let { Φ t} be a non-elliptic continuous semigroup of analytic functions in Ω. Then there exists a univalent function h: Ω →Csuch that h◦Φt(z) = h(z) + t, z ∈Ω, t > 0.(1.3) The function his unique up to an additive constant. The function h is known as the Koenigs function of the semigroup. The interest about such function is that its study can provide quite useful information about the semigroup { Φ t} . Indeed, since the function h is univalent, we can recover {Φt}as Φt(z) = h−1(h(z) + t), z ∈Ω. In fact, the Koenigs function h of the semigroup is linked to the infinitesimal generator H . Let us see this fact. If we differentiate with respect to t in (1.3) and evaluate at t= 0, the chain rule gives in the left-hand side ∂ ∂t(h◦Φt(z))t=0 =h′(Φ0(z))H(s) = h′(z)H(z). where H is the infinitesimal generator of the semigroup { Φ t} . Now, a derivation in the right hand-side of (1.3) gives ∂ ∂t(h(z) + t)t=0 = 1. Putting all together, we find that h′(z) = 1 H(z).(1.4) 6 1.4. The Koenigs function Remark 1.4.2.Even though [ 35 , Theorem 9.3.5] is stated for continuous semigroups in D , the result still holds for continuous semigroups in the right half-plane C+ . This is, given a continuous semigroup { Φ t} in C+ , we consider the conjugated continuous semigroup in D Ψt(z) = L−1◦Φt◦L(z), where L : D→C+ is the Cayley transform L ( z ) = 1+z 1−z . Then, applying [ 35 , Theorem 9.3.5] to the semigroup { Ψ t} and rewriting the result in the the right half-plane, Theorem 1.4.1 follows (see [52, p. 114]). In the case of the right half-plane we will only deal with non-elliptic continuous semigroups and, in particular, those whose Denjoy-Wolff point is ∞ . Nonetheless, let us see how (1.4) looks like for continuous elliptic semigroups in D. First of all, the analogue of Theorem 1.4.1 now takes the form: Theorem 1.4.3. ([ 35 , Theorem 9.3.5]) Let { Φ t} be an elliptic continuous semigroup of analytic functions in D with spectral value λ , λ∈C , Re ( λ ) ≥ 0. Then there exists a univalent function h:D→Csuch that h◦Φt(z) = e−λth(z).(1.5) The function his unique up to a multiplicative constant. Hence, a derivation with respect to ton both sides of (1.5) gives h′(Φt(z)) ∂ ∂tΦt(z) = −λe−λth(z). Evaluating at t= 0, we find that H(z) = −λh(z) h′(z), z ∈D.(1.6) Definition 1.4.4. Let λ∈C\{0}be such that Re(λ)>0,Ωa domain in C. a) We say that Ωis λ -spirallike if e−λt Ω ⊂ Ωfor all t≥ 0. If λ∈ (0 ,∞ ), we say that Ωis spirallike. b) A map h:D→C is λ -spirallike with respect to τ∈D if it is univalent, h ( τ ) = 0 and h ( D )is a λ -spirallike domain. If a map h:D→C is λ-spirallike with λ∈(0,∞), it is also called starlike. There exists a one-to-one correspondence between maps which are spirallike and Koenigs functions of elliptic semigroups in D: Theorem 1.4.5. ([ 35 , Theorem 9.4.3]) Let { Φ t} be an elliptic continuous semigroup in D with Denjoy-Wolff point τ and spectral value λ∈C\ { 0 } with Re ( λ ) ≥ 0. Let h : D→C be its Koenigs function. Then, h is λ -spirallike with respect to τ∈D. Conversely, if h : D→C is λ -spirallike with respect to τ∈D , for some λ∈C\{0}with Re(λ)≥0, then {Φt}where Φt(z) := h−1(e−λth(z)), z ∈D, is an elliptic semigroup in Dwith Denjoy-Wolff point τand spectral value λ. 7 1. Continuous semigroups of analytic functions Definition 1.4.6. Let Ωbe a domain in Cand let Ω0be either Dor C+. a) Ωis said to be starlike at infinity if Ω + t⊆Ωfor all t≥0. b) A map h : Ω 0→C is starlike at infinity with respect to σ∈∂ Ω 0 if it is univalent, h(D)is starlike at infinity and lim supz→σReh(z)=+∞. As in the case of elliptic semigroups, the same correspondence holds between the Koenigs functions of non-elliptic semigroups and the maps which are starlike at infinity: Theorem 1.4.7. ([35, Theorem 9.4.10]) Let h:D→Cbe the Koenigs function of a non-elliptic semigroup { Φ t} in D with Denjoy-Wolff point τ∈∂D . Then, his starlike at infinity with respect to τ. Conversely, given h : D→C starlike at infinity with respect to τ∈∂D and setting Φ t ( z ) = h−1 ( h ( z ) + t )for t≥ 0, then { Φ t} is a non-elliptic continuous semigroup in Dwith Denjoy-Wolff point τ. 1.5 On the uniform convergence of continuous semigroups in C+ The quantitative version of Gumenyuk’s theorem on the uniform convergence of continuous semigroups in the unit disc D relies on a quantitative version of the convergence of continuous semigroups in C+ whose Denjoy-Wolff point is ∞ . This section is devoted to establish this result (Theorem 1.5.13 and Theorem 1.5.18). In the first subsections we present all the ingredients needed for the proof. The section concludes with the the proof of both Theorem 1.5.13 and Theorem 1.5.18. 1.5.1 Preliminary tool: harmonic measure We shall give a brief motivation of the harmonic measure. The reader is addressed to [ 77 , Chapter 4, Section 3] for a more detailed exposition on the topic. The classical Dirichlet problem on the unit disc D consists on finding a harmonic function u on D such that u = ϕ on ∂D , where ϕ is a real continuous function on ∂D . However, we could consider more general domains such as the punctured unit disc, and, even more generally, a domain Ωof the complex plane with non-polar boundary. This is, a domain whose boundary has positive logarithmic capacity. Similarly, we could allow non-continuous initial data ϕ . These considerations give rise to the known as generalised Dirichlet problem. That is, given a domain Ωin C with non-polar boundary and ϕ:∂ Ω →R continuous nearly everywhere (in every point except on a set of zero logarithmic capacity), find the unique harmonic function u on Ωsuch that u ( z ) →ϕ ( ζ ) as z→ζ for nearly every ζ∈∂ Ω(for every ζ∈∂ Ωexcepting a set of zero logarithmic capacity). The harmonic function seeked happens to be the Perron function HΩϕ of the function ϕ on Ω, whose existence is guaranteed under the conditions of the 8 1.5. On the uniform convergence of continuous semigroups in C+ generalised Dirichlet problem. In the classical setting Ω = D and ϕ a Borel bounded function on ∂D,HDϕ=PDϕ, where PDϕis the Poisson integral of ϕ. Now, given a domain Ωwith non-polar boundary and E a Borel subset of ∂∞ Ω, the harmonic measure of E with respect to Ωhappens to be the Perron solution u of the generalised Dirichlet problem for the Laplacian in Ωfor the boundary data χE(z) = (1if z∈E, 0if z∈ E. We will use the standard notation u(z) = ωΩ(z, E), z ∈Ω. We formalise these facts in the next definition. Definition 1.5.1. Let Ωbe a domain in C and denote by B ( ∂ Ω) the Borel σ - algebra of ∂ Ω. A harmonic measure for Ωis a function ωΩ : Ω ×B ( ∂ Ω) → [0 , 1] such that: •for all z∈Ω,ωΩ(z, ·)is a Borel probability measure on ∂Ω; •if ϕ:∂Ω→Ris continuous, then HΩϕ=PΩϕ, where PΩϕ(z) := Z∂Ω ϕ(ζ)·ωΩ(z, dζ) is the generalized Poisson integral of ϕon Ω. There is a quite ilustrative way of seeing harmonic measures. Indeed, consider Ω ⊂C a domain with non-polar boundary. Suppose z∈ Ωand B∈ B ( ∂ Ω). We denote by Bt , t > 0, the Brownian motion in C starting from z and set t0 = inf{t > 0 : Bt∈ Ω } , this is, the first exit time from Ω. Then, the harmonic ωΩ(z, B)is the probability that Bt0∈B. Figure 1.1: The harmonic measure as a Brownian motion. 9 1. Continuous semigroups of analytic functions Nonetheless, we will not use in the thesis this interpretation in terms of brownian motion. Given E⊆C , we denote by ℓ ( E )the outer linear measure of E (see [ 74 , page 129]). If Eis a Jordan arc, then ℓ(E)is nothing but its length. Example 1.5.2. (See [77, p. 96]) In the case Ω = D, we have that ωD(z, dζ) = 1−|z|2 |z−ζ|2|dζ| 2π, where |dζ| is the arc-length measure. Therefore, given B a Borel subset of T , Definition 1.5.1 gives ωD(z, B) = PΩχB(z) = ZB ωD(z, dζ) = ZB 1−|z|2 |z−ζ|2|dζ| 2π·(1.7) Hence, if z= 0, then ωD(0, B) = ZB 1 |ζ||dζ|=1 2πℓ(B). This is, for z = 0, the harmonic measure in D of a Borel set B at the point z is the normalised arc-length measure of the set B. In our setting, the domains shall all be simply connected. This case is contained in the more general previous setting, as the boundary of a simply connected domain Ωcontains a continuum. Since every continuum is a non-polar set, [ 77 , Corollary 3.8.5], harmonic measures are defined for simply connected domains Ω. In this more straightforward case, a good reference for the properties we will use is [ 74 ] (see also [ 35 ]). In fact, the simply connected case, following an idea of Pommerenke [ 74 ], can be treated more easily by defining the harmonic measure in the unit disc as ωD(z, B) := PDχB(z) = 1 2πZ2π 0 1−|z|2 |eiθ−z|χB(eiθ)dθ, z ∈D, B ∈∂D,(1.8) and then carrying it to any simply connected domain Ωusing Riemann maps. More precisely, given a simply connected domain Ωin C ,Ω  = C , and f : D→ Ω a Riemann map of the domain Ω, we can define its harmonic measure in terms of the harmonic measure in the unit disc Dfrom (1.8) as ωΩ(z, B) := ωD(f−1(z), f−1(B)), z ∈Ω, B ∈ B(∂Ω),(1.9) where f−1 ( B ) = {ξ∈∂D : ∠limz→ξf ( z ) ∈B} . The measure ωΩ ( z, · )does not depend on the conformal representation f chosen. This is a consequence of the invariance of the harmonic measure in the unit disc D under automorphisms. Lemma 1.5.3. ([35, Proposition 7.1.4 (4)]) Let T∈Aut(D). Then, ωD(z, B) = ωD(T(z), T(B)),for all z∈D. Lemma 1.5.4. Let Ωbe a simply connected domain in C . Then, the harmonic measure defined in (1.9) does not depend on the Riemann function of the domain chosen. 10 1.5. On the uniform convergence of continuous semigroups in C+ Proof. Let f1, f2 : D→ Ωbe to Riemann maps of Ω. Let T∈Aut ( D )so that f1=f2◦T. Set ω1 Ω(z, B) := ωD(f−1 1(z), f−1 1(B)), ω2 Ω(z, B) := ωD(f−1 2(z), f−1 2(B)). Now, by Lemma 1.5.3, ω1 Ω(z, B) = ωD(T−1(f−1 2(z)), T−1(f−1 2(B))) =ωD(f−1 2(z), f−1 2(B)) =ω2 Ω(z, B). ■ See [ 35 , Section 7.2] for explicit computations for some simply connected domains Ω. Lemma 1.5.3 falls in the following general case. Proposition 1.5.5. ([ 77 , Theorem 4.3.8]) Let Ω ⊂C be a domain with non-polar boundary and B∈ B(∂Ω). Consider fa conformal mapping on Ω. Then, ωΩ(z, B) = ωf(Ω)(f(z), f(B)), z ∈Ω. We state the last property of harmonic measures that we will need later. It is sometimes known as the subordination principle. Proposition 1.5.6. ([ 77 , Corollary 4.3.9]) Let Ω 1, Ω 2⊂C be two domains with non-polar boundary such that Ω1⊂Ω2and B∈ B(∂Ω1)∩B(∂Ω2), then ωΩ1(z, B)≤ωΩ2(z, B), z ∈Ω1. 1.5.2 Preliminary tool: Lavrentiev’s Theorem Theorem 1.5.7 (Lavrentiev).([ 69 ], see also [ 74 , Proposition 6.11] and [ 66 , Theorem 2.1]) Let Ωbe a simply connected region in the complex plane such that Ω ⊃D and ∂ Ωis a Jordan curve with L = ℓ ( ∂ Ω) <∞ . Consider f : D→ Ωa conformal representation satisfying f (0) = 0. Then, for every ε > 0, there exists δ=δ(ε, L)>0such that for every A⊂∂Ω, if ℓ(A)≤δ, then ℓ(f−1(A)) < ε. For our purposes, it is more convenient to rewrite Lavrentiev’s theorem in terms of harmonic measures. In order to do that, let f : D→ Ωbe a conformal representation satisfying f (0) = 0 and A⊂∂ Ω. By Carathéodory’s theorem (see [ 35 , Theorem 4.3.3] or [ 74 , Theorem 2.6]), since ∂ Ωis a Jordan curve, f extends to a homeomorphism from Dto Ω. By (1.9), it follows that ωD(0, f−1(A)) = ωΩ(0, A). That is, ℓ ( f−1 ( A )) = ωΩ (0 , A ). Taking this into account, we have the following immediate consequence of Lavrentiev’s Theorem. Corollary 1.5.8. There exists a constant ρ > 0such that the following holds: Let a∈ (0 , + ∞ ). Consider a simply connected domain Ωin C such that ∂ Ωis a Jordan curve with ℓ ( ∂ Ω) ≤ 4 a and D ( w, a/ 4) ⊂ Ωfor some w∈ Ω. Then, ωΩ(w, A)<1 8, whenever Ais a measurable subset of ∂Ωwith ℓ(A)≤ρa. 11 1. Continuous semigroups of analytic functions Proof. Take ρ = δ (1 / 8 , 16) / 4the constant provided by Theorem 1.5.7. We may assume that w = 0. Call b Ω = 4 a Ω. Notice that D⊂b Ω and ℓ ( ∂b Ω ) ≤ 16. Let A be a measurable subset of ∂ Ωwith ℓ ( A ) ≤ρa . Consider b A = 4 aA⊂∂b Ω . Then ℓ(b A)≤4ρ=δ(1/8,16). Therefore, ωΩ(0, A) = ωb Ω(0,b A)<1/8. ■ The key of the corollary is that the control on ωΩ ( w, A )depends on a but not on the the conformal representation of the domain Ω, being the control uniform for every domain such that ∂Ωis a rectifiable Jordan curve. 1.5.3 The main theorem Let θ≥ 0. The space H∞ ( Cθ )consists on the bounded analytic functions in Cθ. We will write H∞(C+)instead of H∞(C0). The growth rate of the modulus of the infinitesimal generator of a continuous semigroup close to the imaginary axis is intimately related to the rate of convergence of the Re (Φ t ( z )) to Re ( z )as t goes to zero and, as we are about to see in Theorem 1.5.13, it is also deeply linked to the ratio of convergence of {Φt}to the identity map. Before moving on, let us point out a fact which will be frequently used in this section and, in particular, in the forthcoming lemma. Remark 1.5.9.Let H : C+→C+ be the infinitesimal generator of a continuous semigroup {Φt}in C+. Then, by Theorem 1.3.1, we have that d dt Re(Φt(z)) = ReH(Φt(z)) ≥0,for all z∈C+. Therefore, the map t7→ Re(Φt(z)) is increasing. Lemma 1.5.10. Let 0 ≤α≤ 1. Let H : C+→C+ be the infinitesimal generator of a continuous semigroup { Φ t} in C+ such that there exist a constant K > 0 and ε0>0satisfying sup z∈Cε|H(z)| ≤ K εα,for all 0< ε < ε0. Then, there exists B = B ( K ) > 0and e t0 = e t0 ( ε0 ) > 0such that if t < ˜ t0 , we have |Re(Φt(z)) −Re(z)| ≤ Bt 1 1+α,for all z∈C+. Proof. Fix 0 < t < ε2 0 . We can assume that ε0< 1. Take ε = t1 1+α> 0and choose z∈C+ . If Re (Φ t ( z )) < ε , we are done, because Re (Φ t ( z )) −Re ( z ) < ε = t1 1+α . Assume then the existence of 0 ≤t1 = t1 ( z ) < t such that Re (Φ u ( z )) ≥ε , for all u≥t1 and Re (Φ u ( z )) ≤ε for every u<t1 . Such t1 exists thanks to the fact that u7→ Re (Φ u ( z )) is non-decreasing and continuous. Observe that t1 could be 0and, in this case, Φt1(z)−z= 0. Now, Re(Φt(z)) −Re(z) = Zt 0 Re(H(Φu(z)))du 12 1.5. On the uniform convergence of continuous semigroups in C+ =Zt1 0 Re(H(Φu(z)))du +Zt t1 Re(H(Φu(z)))du. Using the hypothesis on H , the definition of t1 and, the choice of ε , we find that Zt1 0 Re(H(Φu(z)))du +Zt t1 Re(H(Φu(z)))du ≤Re(Φt1(z)−z) + K εαt ≤ε+K εαt= (1 + K)t1 1+α, and the conclusion follows. ■ Lemma 1.5.11. Let f : [ c, d ] ⊂R→R be a C1 function of Lipschitz constant K. Consider g: [c, d]→Rgiven by g(x) = min{f(y) : y≥x}. Then, 1) The function gis Lipschitz with constant K. 2) If g(x)< f(x), then g′(x)=0. 3) If g′(x)exists and g′(x)>0, then g′(x) = f′(x). Proof. First, notice that g′ ( x ) ≥ 0, whenever such a value does exist, and g ( x ) ≤f ( x ), for all x . For the first statement, consider x1, x2∈ [ c, d ]and, without loss of generality, we assume x1< x2 . Take yj≥xj such that f ( yj ) = g ( xj ), for j = 1 , 2. On the one hand, if y1≥x2 , then y1 = y2 and g(x1) = g(x2). On the other hand, if y1< x2, then |g(x2)−g(x1)|=g(x2)−g(x1) = f(y2)−f(y1)≤f(x2)−f(y1) ≤K(x2−y1) ≤K(x2−x1). So that gis Lipschitz with constant K. For part b), notice that if g ( x ) < f ( x ), by continuity, there exists δ > 0 such that sup y∈(x−δ,x+δ) g(y)<inf y∈(x−δ,x+δ)f(y). However, by the definition of g , this implies that g is constant in such an interval and, therefore, g′(x)=0. Consider x a point such that g′ ( x )exists and g′ ( x ) > 0. Then, by b), we have that g ( x ) ≥f ( x ). However, by the definition of g , g ( x ) ≤f ( x ). Therefore, f ( x ) = g ( x ). Now, for some δ > 0small enough, either f ( y ) > g ( y )for all y∈Iδ = [ x, x + δ ), either there exists a sequence {yn} converging to x , yn> x for all n and such that f ( yn ) = g ( yn )for all n . If the second case holds, the conclusion follows since f′(x) = lim y→x+ f(y)−f(x) y−x= lim n→∞ f(yn)−f(x) yn−x= lim n→∞ g(yn)−g(x) yn−x=g′(x). Therefore, let us assume that the first case occurs. Then, by the argument to prove statement b), we would have that g is constant in Iδ . Therefore, the right-hand derivative of g at the point x would be zero. Nonetheless, as g′ ( x ) exists, the left-hand derivative would also be zero, meaning that g′ ( x )=0, which contradicts the initial assumption on x.■ 13 1. Continuous semigroups of analytic functions Remark 1.5.12.According to Theorem 1.3.4, the continuous semigroups { Φ t} in the right half-plane with Denjoy-Wolff point ∞ are those whose infinitesimal generator H is a holomorphic map in C+ and such that H : C+→C+ . The case in which the infinitesimal generator H touches the boundary of C+ , this is, H ( C+ ) ∩∂C+ = ∅ , corresponds to the semigroup { Φ t} given by Φ t ( z ) = z + iat , a∈R , t > 0. The statements of Theorem 1.5.13 and Theorem 1.5.18 clearly hold for this case. Because of this, we shall exclude the case in which H touches ∂C+in the statments of the theorems. Theorem 1.5.13. Let 0 ≤α≤ 1. Let H : C+→C+ be a holomorphic function and denote by { Φ t} the associated continuous semigroup. Suppose that there exist a constant K > 0and ε0>0so that sup z∈Cε|H(z)| ≤ K εα,for all 0< ε < ε0. Then, there exist a constant A = A ( ε0, α, K )and a t0 = t0 ( ε0, K ) > 0such that if t < t0, we have |Φt(z)−z| ≤ At 1 1+α,for all z∈C+. Proof. Step I: Simplifications. By Lemma 1.5.10, there exist B = B ( K ) > 0and e t0 = e t0 ( ε ) > 0such that if t < e t0, then |Re(Φt(z)) −Re(z)| ≤ Bt 1 1+α,for all z∈C+.(1.10) Therefore, it is enough to show that there exist a constant e A and t0> 0such that if t<t0, then |Im(Φt(z)) −Im(z)| ≤ e At 1 1+α,for all z∈C+. Fix z∈C+ and 0 <t<e t0 . Let ρ be the universal constant provided in Corollary 1.5.8 and take C1>max{ 4 , 2 /ρ} . Write a = |Im (Φ t ( z )) −Im ( z ) | . Assume for the moment that a≤2ε0.We may suppose that Re(Φt(z)) −Re(z)≤1 C1|Im(Φt(z)) −Im(z)|,(1.11) otherwise taking ˜ A > BC1 we are done. We shall carry out the proof for the case Im (Φ t ( z )) −Im ( z ) = a . The prove of the other case is done in a similar fashion. Define the curve γ: [0,t]→C+ u7→ Φu(z). We denote by γ∗ the image of the interval [0 , t ]under γ . Similarly, we consider the square C={s∈C+:Re(z)<Re(s)<Re(z) + a, Im(z)<Im(s)<Im(z) + a} and set w to be the centre of the square C . In fact, we can assume that the whole γ∗lies in C. Indeed, define t1= sup{u∈(0, t) : Im(Φu(z)) <Im(z)}, 14 1.5. On the uniform convergence of continuous semigroups in C+ t2= inf{u∈(t1, t) : Im(Φu(z)) >Im(z) + a}. Then, for all τ∈[t1, t2], Im(z)≤Im(Φτ(z)) ≤Im(z) + a. Let z0= Φt1(z)and υ=t2−t1. By the semigroup structure, Φt2(z)=Φυ(z0). Then, a=Im(Φt2(z)) −Im(Φt1(z)) = Im(Φt2−t1(Φt1(z))) −Im(Φt1(z)) =Im(Φυ(z0)) −Im(z0). On the other hand, a=Im(Φt(z)) −Im(z). Hence, if we prove that Im (Φ υ ( z0 )) −Im ( z0 ) ≤A√υ , the result will follow since Im(Φt(z)) −Im(z) = Im(Φυ(z0)) −Im(z0)≤A√υ≤A√t. Hence, we assume that γ∗ is in C . In order to ease the reading of the proof, we define δ := Re (Φ t ( z )) −Re ( z ). With this notation, (1.10) becomes δ≤Bt 1 1+α . By (1.11) and the choice of C1we have that δ≤a/C1< a/4.(1.12) Also, clearly, Re(w)> a/2. Therefore, by hypothesis, |H(w)| ≤ 2K aα·(1.13) Step II: Construction of Ω. We are going to consider the curve α : [0 , t ] →C+ given by α ( u ) = f ( u ) + ig ( u ), where f ( u ) = Re ( γ ( u )) and g ( u ) = min{Im ( γ ( s )) : s≥u} which satisfies the thesis from Lemma 1.5.11, so the imaginary part of the curve α is non-decreasing. We shall also denote by α∗ to α ([0 , t ]). The curve α∗ divides the square C into two regions. Let Ωbe the right hand-side region of the square (see Figure 1.2). By the definition of δ , since δ < a/ 4, we have that Re ( w ) > δ + a 4 and, consequently, D(w, a/4) ⊂Ω. z Φt(z) γ∗ Im (z) Im (z) + a Re (z) γ∗ nα∗ z Φt(z) Ω w Im (z) Im (z) + a Re (z) Re (z) + a Figure 1.2: The domain Ωand its boundary. 15 1. Continuous semigroups of analytic functions 1.6 On the uniform convergence of continuous semigroups in D The first result regarding the uniform convergence of continuous semigroups in the whole unit disc D appeared in 2005 in Contreras and Díaz-Madrigal’s work [ 43 ]. There, having as a starting point the study of the semigroups of composition operators in the disc algebra, the authors proved that whenever the functions of a continuous semigroup { Φ t} in D have a continuous extension to the boundary of the unit disc for all t > 0, then sup z∈D|Φt(z)−z| → 0,as t→0+.(1.18) Years later, in 2014, P. Gumenyuk, using an argument involving no Koebe Arc’s Theorem ([ 35 , Theorem 3.2.4]), was able to remove the restriction of the extension to the boundary of the elements of the semigroup leading to the following striking result: Theorem 1.6.1. [ 62 , Proposition 3.2] Let { Φ t} be a continuous semigroup of analytic functions in the unit disc D . Then, the semigroup converges uniformly in Dto the identity map as tgoes to zero. In this section, we will show how using Theorem 1.5.18 and the ideas involved in its proof, we can provide the following quantitative version of Gumenyuk’s result. Theorem 1.6.2. Let { Φ t} be a continuous semigroup of analytic functions in the unit disc D. Then, there exists C > 0such that |Φt(z)−z| ≤ C√t, (1.19) for all z∈Dand for all t > 0. Before moving on to the proof of the theorem, it is worth pointing out that this result is far from being true for continuous semigroups in the right half-plane, as it was shown in Example 1.5.19. We will split the proof of Theorem 1.6.2 in two parts. First, we shall begin by proving Theorem 1.6.2 for elliptic semigroups. Then, we will carry out the proof for the non-elliptic case. The following auxiliary lemma will be needed in both proofs, so we prove it right straight-away. The first statement is well-known and the reader may find a proof of it in any Complex Analysis manual. Lemma 1.6.3. Let p : D→C+ be an analytic function, G ( z ) = ( z− 1) 2p ( z ), z∈D , and G1 : C+→C+ given by G1 ( w )=2 p (( w− 1) / ( w + 1)), w∈C+ . Then, there exists a constant M > 0such that the following statements hold: a) For any z∈D, |p(z)| ≤ M 1−|z|2· b) Let λ∈(0,1) and consider z∈Hλ:= D(λ, 1−λ). Then |G(z)| ≤ M λ· 22 1.6. On the uniform convergence of continuous semigroups in D c) If w=x+iy ∈C+, then |G1(w)| ≤ M(1 + x)2+y2 2x· Proof. Part a )is a well-known result which follows from Herglotz’s representation formula. Regarding statement b ), take z∈D ( λ, 1 −λ ). Then there exists β∈ ( λ, 1) such that |z−β| = 1 −β . That is, z = (1 −β ) eiθ + β for some θ∈ (0 , 2 π ). A quick computation shows |z−1|2 1−|z|2=(1 −β)2|eiθ −1|2 β(1 −β)|eiθ −1|2=1−β β· Therefore, by a), and the fact that β > λ |G(z)|=|z−1|2|p(z)|≤|z−1|2M 1−|z|2=M1−β β≤M1 β≤M1 λ· Claim c)also follows from a). Indeed, |G1(w)|= 2pw−1 w+ 1≤2M|w+ 1|2 |w+ 1|2−|w−1|2 = 2M(1 + x)2+y2 4x· ■ 1.6.1 Proof of the elliptic case Proof of the elliptic case of Theorem 1.6.2. Let { Φ t} be an elliptic continuous semigroup on D . Then, the Denjoy-Wolff point τ of the semigroup { Φ t} lies in D . By a standard conjugation argument, we may assume that τ = 0. To see this, consider the Möbius transform T : D→D given by T ( z ) = z+τ 1+τz . Then, if we set Ψ t ( z ) = T−1◦ Φ t◦T ( z ), clearly, { Ψ t} is a continuous semigroup on D and it satisfies that DW (Ψ t )=0. Moreover, since T is K -Lipschitz, if we prove that | Ψ t ( z ) −z| ≤ C1√t , then, the semigrop { Φ t} would satisfy (1.19) with C=C1K. Indeed, setting w=T(z), we would have that C1√t≥ |Ψt(z)−z| ≥ K−1|Φt(w)−w|. Therefore, we suppose τ= 0. Observe also that, since | Φ t ( z ) −z| ≤ 2, it is enough to get the thesis if t is close to zero. By the Schwarz Lemma, we have that for every t > 0, | Φ t ( z ) |≤|z| . This, together with the algebraic structure of the semigroup, gives us that | Φ t ( z ) | increases to |z|, as tdecreases to 0, for all z∈D. We shall denote by G the infinitesimal generator of the semigroup { Φ t} . It is known that G ( z ) = −zp ( z )(see Theorem 1.3.3), where p : D→C+ is analytic. Set C1:= supu∈D(0,1/2) |G(u)|. Take z0∈Dand define t1=t1(z0) = inf{t≥0 : |Φt(z0)| ≤ 1/2}. 23 1. Continuous semigroups of analytic functions Notice that t1is 0if |z0| ≤ 1/2. Then, by the definition of t1, we have that |Φt(z0)−z0|=Zt 0 G(Φτ(z0))dτ≤Zt1 0|G(Φτ(z0))|dτ +Zt t1|G(Φτ(z0))|dτ ≤Zt1 0|G(Φτ(z0))|dτ +tC1. Therefore, we can suppose that |z0|> 1 / 2and it suffices to bound the first term in the latter inequality. Knowing this, from now on, fix z0 = reiθ0 with r∈ (1 / 2 , 1) and θ0∈ [0 , 2 π ]. Consider the set A={z∈D:1 2<|z|<1,−π/2 + θ0<arg(z)< π/2 + θ0}. Observe that inf{|z−z0|:|z|>1/2, z /∈ A} ≥ √2/2. Claim. There exists C = C ( M ) > 0such that if Φ u ( z0 ) ∈ A for all 0 ≤u<t then |Φt(z0)−z0| ≤ C√t. Let us see that this is enough to conclude the proof and then we will prove the claim. Take t2≤t1 such that Φ t2 ( z0 ) ∈∂A and Φ u ( z0 ) ∈ A for all u < t2 . If t2 = t1 we conclude the proof by using the claim. If t2< t1 , then |arg (Φ t2 ( z0 )) −θ0| = π/ 2and | Φ t2 ( z0 ) −z0| ≥ √2/ 2. Moreover, |Φt2(z0)−z0| ≤ C√t2, giving √2≤2C√t2. Thus, for t2< t ≤t1we have |Φt(z0)−z0| ≤ 2≤2√2C√t2<2√2C√t. And the proof would be finished with the constant 2√2C. Proof of the Claim. Let T ( z ) = −log ( z ), where log denotes the branch in A of the logarithm such that log ( z0 ) = ln|z0| + iθ0 , and set w0 = −log ( z0 ). Notice that R:= T(A) = {w∈C+: 0 <Re(w)<log 2,−π/2−θ0<Im(w)< π/2−θ0}. Define p1 : C+→C+ by p1 ( w ) = p ( e−w ). Hence, it is the infinitesimal generator of a continuous semigroup in C+ (see Theorem 1.3.4). Let us denote such a semigroup by { Ψ t} . Furthermore, the generator p1 satisfies statement 1) in Theorem 1.5.18. Indeed, sup w∈C1/2|p1(w)|= sup w∈C1/2|p(e−w)|= sup |z|≤e−1/2|p(z)|<∞. Consider the curve γ : [0 , t ) → R given by γ ( u ) = −log (Φ u ( z0 )). Then a simple computation yields γ′(u) = −G(Φu(z0)) Φu(z0)=p(Φu(z0)) = p(exp(−γ(u))) = p1(γ(u)). This means that Ψ u ( w0 ) = γ ( u ). Therefore, applying Theorem 1.5.18 [(i) implies (iv)] and using the fact that the function e−w is 1-Lipschitz in C+ (its derivative is bounded by 1in C+ ), there is a constant A = A ( C1 )and t0 such that A√t≥ |Ψt(w0)−w0|=|log(Φt(z0)) −log(z0)|≥|Φt(z0)−z0|, whenever t<t0, and the claim follows. ■ 24 1.6. On the uniform convergence of continuous semigroups in D 1.6.2 Proof of the non-elliptic case Proof of the non-elliptic case of Theorem 1.6.2. Take a non-elliptic continuous semigroup { Φ t} . We may assume that its Denjoy-Wolff point is 1and again it is enough to get the thesis when t is close to zero. Let us call H the infinitesimal generator of the semigroup. By the Berkson-Porta decomposition (see Theorem 1.3.3), there is p:D→C+holomorphic such that H(z)=(z−1)2p(z).(1.20) Moreover, by Lemma 1.6.3, there is M > 0such that, for all 0<λ<1 |H(z)| ≤ M λ, z ∈D(λ, 1−λ). Take λ = √t≤ 1 / 2and z∈D . We recall that, by Julia’s Lemma [ 35 , Theorem 1.4.7], Hλ = D ( λ, 1 −λ )is invariant under the elements of the semigroup. Write t0 = t0 ( z ) = inf{u : Φ u ( z ) ∈Hλ} . Observe that the trajectory could lie outside the horodisc Hλ. In such case, t0= +∞. If t0< t, then |Φt(z)−Φt0(z)| ≤ Zt t0|H(Φτ(z))|dτ ≤Mt−t0 λ=Mt−t0 √t· If we prove that there is C such that | Φ u ( z ) −z| ≤ C√u for all u<t0 , we are clearly done in the case t≤t0 . Moreover, in the case that t0< t , we also have that |Φt(z)−z| ≤ max {M, C}(√t0+t−t0 √t)≤2 max {M, C}√t and the proof would be concluded. This means that we may assume that t<t0 , that is, we assume that the trajectory y ( u )=Φ u ( z ),0 ≤u≤t lies outside the horodisc Hλ. Let T : D→C+ , T ( s ) = 1+s 1−s , s∈D . We set A = T ( z )and B = T (Φ t ( z )). Notice that T ( Hλ ) = Cλ 1−λ . Moreover, since we took λ≤ 1 / 2, we also have that Cλ 1−λ⊃C2λ. Applying again Julia’s Lemma, the map u7→ Re(T(Φu(z))) is non-decreasing. We know that Re ( B ) −Re ( A ) ≤ 2 λ because the curve T (Φ t ( z )) does not touch Cλ 1−λ . Call d = |Im ( A ) −Im ( B ) | . We also may assume that Im ( A ) ≤Im ( T (Φ u ( z ))) ≤Im ( A ) + d = Im ( B ), for all 0 ≤u≤t . This simplification is the same as the one done in the proof of Theorem 1.5.13 on page 14. Let G1 ( w )=2 p ( T−1 ( w )), w∈C+ . Set γ : [0 , t ) →C+ , where γ ( u ) = T ( y ( u )). G1 is the infinitesimal generator of a semigroup in the right half-plane C+(see Theorem 1.3.4) and γ′(u) = T′(y(u))y′(u) = 2 (1 −y(u))2 ∂ ∂u(Φu(z)) = 2 (1 −y(u))2H(y(u)) = 2p(y(u)) = G1(γ(u)), for all u > 0, where in the last identity we have used (1.20) . Thus, γ is the trajectory of the continuous semigroup generated by G1 . Notice that γ∗ = γ ([0 , t ]) lies in the vertical strip Ω 2 = {s∈C+ : Re ( s ) < 2 λ} . Now, since T−1(w) = w−1 w+1 , a simple computation shows that |Φt(z)−z|=|T−1(A)−T−1(B)|= 2 |A−B| |1 + A||1 + B|· 25 1. Continuous semigroups of analytic functions Denote by ρ the constant provided by Corollary 1.5.8 and M the constant associated with p given in Lemma 1.6.3. Take η≥max{ 18 , 24 /ρ, 25 M 384 /ρ} . If | Φ t ( z ) −z| ≤ η√t , then we are done. Assuming on the contrary that |Φt(z)−z|> η√t, we will get a contradiction. Notice that in such a case |A−B|>1 2η√t|1 + A||1 + B| ≥ √t 2η=λη 2,(1.21) where in the identity we are using that λ = √t . Hence, recalling that η≥ 18 and Re(A)−Re(B)≤2λ, from (1.21) we obtain that d2>λ2 4η2−(Re(A)−Re(B))2≥λ2 4η2−4λ2=λ2(η2/4−4) ≥64λ2. That is, d > 8λ≥4(Re(B)−Re(A)).(1.22) Once again, we argue as in the proof of Theorem 1.5.13. Indeed, consider the square C={w∈C+:Re(A)<Re(w)<Re(A) + d, Im(A)<Im(w)<Im(A) + d}. Let w0 = A + (1 + i ) d/ 2. Recall that γ : [0 , t ] → Ω 2∩C , with γ ( u ) = T ( y ( u )). Similarly, we define the curve α : [0 , t ] →C+ given by α ( u ) = f ( u ) + ig ( u ), where f ( u ) = Re ( γ ( u )) and g ( u ) = min{Im ( γ ( x )) : u≤x≤t} . We shall also denote by γ∗ the set γ ([0 , t ]) and by α∗ the set α ([0 , t ]). The curve α∗ divides the square C into two regions. Let Ωbe the right hand-side region of the square (see Figure 1.2 on page 15). Now, we introduce the sets A0={s∈α∗:s∈ γ∗},A1=s∈α∗∩γ∗:|G1(s)| ≤ ρ 3|A−B| t and set A = A0∪A1 . For E⊂∂ Ω, ω ( E ) = ωΩ ( w0, E ). Arguing as in the proof of Theorem 1.5.13, we get that ω ( α∗ ) ≥ 1 / 4. We now estimate ℓ ( A )in order to apply Corollary 1.5.8. Using that α∗ ([0 , t ]) lies in the vertical strip Ω 2 , the definition of A1as well as Lemma 1.5.11, we obtain ℓ(A)≤Zt 0 Re α′(u)du +Zα−1(A1) Im α′(u)du ≤2λ+ρ 3|A−B|. Using (1.21) first and then (1.22), 2λ+ρ 3|A−B| ≤ 4 η+ρ 3|A−B| ≤ 24 η+ρ 3d. Since η > 24 /ρ , then ℓ ( A ) ≤ρd . Of course, D ( w0, d/ 4) ⊂ Ω. By Corollary 1.5.8, we have that ω(A)<1/8. Since ω(α∗)≥1/4, we conclude that ω(α∗\A)>1 8·(1.23) By the definition of harmonic measure and the fact that Re ( √G1 )is a positive harmonic function, we get p|G1(w0)| ≥ RepG1(w0) =Z∂Ω Re(pG1(s))dω(s)≥Zα∗\A Re(pG1(s))dω(s). 26 1.6. On the uniform convergence of continuous semigroups in D Using first the fact that Re ( √G1 ) ≥Im ( √G1 ), then the definition of the set A and finally estimate (1.23), Zα∗\A Re(pG1(s))dω(s)≥Zα∗\A p|G1(s)| √2dω(s)≥1 √2rρ 3|A−B| tω(α∗\A) ≥1 √2rρ 3|A−B| t 1 8· Putting all together, we have shown that |G1(w0)| ≥ ρ 384 |A−B| t·(1.24) Let us first assume that | 1 + A| and | 1 + B| are equivalent, namely, let us suppose that 1 / 2 ≤ | 1 + A|| 1 + B|−1≤ 2. Using both the first inequality in (1.21) and (1.24), we have that |G1(w0)|>ρ 384 1 2η1 √t|1 + A||1 + B|. We recall that w0 = x + iy is the centre of the square C . Now, | 1 + A|| 1 + B| ≥ N/ 2, where N = max ( | 1 + A|2,| 1 + B|2 ). Then, since Im ( A ) < y < Im ( B ), we have N≥y2 . The definition of x together with the fact that d > 8 λ (see the first inequality in (1.22)) imply that x < 2λ+d 2≤3 4d < 3 2max(|Im(A)|,|Im(B)|)≤3 2√N. (1.25) In particular, 1 + x≤5 2√N . Using these estimates on x and y , we find that 8N > y2+ (1 + x)2. Therefore, |G1(w0)|>ρ 384 1 32η1 √t(y2+ (1 + x)2)≥ρ 384 1 16ηy2+ (1 + x)2 x, where we have used 2√t= 2λ<x. Finally, Lemma 1.6.3 c) gives |G1(w0)|>ρ 384 1 8Mη|G1(w0)|. Since η > 8 M384 ρ , we obtain a contradiction and, consequently, it cannot occur that 1/2≤ |1 + A||1 + B|−1≤2. Suppose now that | 1 + B|> 2 | 1 + A| . The case 2 | 1 + B|<| 1 + A| is proven in a same fashion. Using Lemma 1.6.3, we have that |G1(w0)| ≤ M(1 + x)2+y2 2x·(1.26) Notice that, by our assumption, |A−B| ≥ |1 + B|−|1 + A| ≥ |1 + B| 2· Therefore, we have x≥d 2≥|1 + B| 8(1.27) 27 1. Continuous semigroups of analytic functions and |y| ≤ |Im A|+|Im B| 2≤|1 + A|+|1 + B| 2<|1 + B|.(1.28) Now, as |1 + B|>2|1 + A|, d=|Im(A)−Im(B)| ≤ |Im(A)|+|Im(B)| ≤ |1 + A|+|1 + B| ≤ 3 2|1 + B|. Eventually, using again the fact that x≤3 4d (arguing as in (1.25) ), we have x≤ 9 | 1 + B|/ 8. In particular, (1 + x ) 2≤ 289 | 1 + B|2/ 64. Using this in (1.26) together with estimates (1.28) and (1.27) , we find that |G1(w0)| ≤ M(1 + x)2+y2 2x<353M 64 |1 + B|2 2x<353M 16 |1 + B|2 <23M|1 + B|. On the other hand, using (1.24), (1.21), and that √t≤1/2, we have |G1(w0)|>ρ 384 1 2η1 √t|1 + A||1 + B| ≥ ρ 384η|1 + A||1 + B| ≥ρ 384η|1 + B|. As ηwas chosen so that η > 25M384 ρ, we reach the desired contradiction. ■ We conclude the section providing an example showing that the statement of Theorem 1.6.2 cannot be strengthened in the sense that t1/2 cannot be replaced by another function that goes to zero faster than t1/2. Proposition 1.6.4. There exists a continuous semigroup { Φ t} of holomorphic functions in Dsuch that lim inf t→0+ supz∈D|Φt(z)−z| √t>0. Proof. Consider p : D→C+ , p ( z ) = (1 + z ) −1 , z∈D , and the infinitesimal generator given by H(z) = (1 −z)2p(z) = (1 −z)2 1 + z, z ∈D. Let { Φ t} be its associated continuous semigroup. Its Denjoy-Wolff point is 1 and since H ( x )is real whenever x∈ ( − 1 , 1), we have that Φ t ( x ) ∈ ( − 1 , 1) for all x∈(−1,1). Consider the vector field G ( z ) = 1 1+z , with z∈ ( − 1 , 0). For x∈ ( − 1 , 0) fixed, the solution of the initial value problem y′(t) = G(y(t)), y(0) = x is given by yx(t) = p2t+ (1 + x)2−1for all t > 0. We claim that for every t > 0and x < 0such that Φ t ( x ) < 0, we necessarily have that Φ t ( x ) ≥yx ( t ). Fix t > 0and x < 0so that Φ t ( x ) < 0. Then, we have 28 1.6. On the uniform convergence of continuous semigroups in D that Φ s ( x ) < 0for all 0 ≤s≤t . Define e ( u ) = yx ( u ) − Φ u ( x ). Observe that e(0) = 0 and e′(u) = ∂ ∂t (yx(t)−Φt(x))t=u=G(yx(u)) −H(Φu(x)).(1.29) Since H ( z ) > G ( z )for all z∈ ( − 1 , 0), we have that e′ (0) < 0. This, together with the fact that e (0) = 0, imply that e ( u ) < 0for u small enough. This is, Φu(x)≥yx(u)for usmall enough. Let us now suppose the existence of u<t such that yx ( u ) > Φ u ( x ). Define t1= inf{u > 0:Φu(x)<0, e(u)=0}. The assumption on u and the fact that e ( w ) < 0for w small enough imply that t1< t. Thus, for u < t1,e(u)<0. Then, e′(t1)≥0. But, e′(t1) = G(yx(t1)) −H(Φt1(x)) <0 since H ( u ) > G ( u )for all u∈ ( − 1 , 0). A contradiction. Therefore, Φ t ( x ) ≥yx ( t ) for all x∈(−1,0) and all tsuch that Φt(x)<0. Notice that the curve t7→ Φ t ( − 1 / 2) is contained in the interval ( − 1 , 1) and limt→∞ Φ t ( − 1 / 2) = 1. Then, we can take t0 such that Φ t0 ( − 1 / 2) = 0. Then Φt(x)<0for all x < −1/2and t<t0. Thus, for t<t0 sup z∈D|Φt(z)−z| ≥ sup x<−1/2|Φt(x)−x|= sup x<−1/2 (Φt(x)−x) ≥sup x<−1/2 (yx(t)−x) = sup x<−1/2 (p2t+ (1 + x)2−1−x). If x < √t/2−1, then p2t+ (1 + x)2−1−x≥√t. Therefore, sup z∈D|Φt(z)−z| ≥ √t, for all t<t0, and the claim follows. ■ Example 1.6.5. The constant A from the elliptic case of Theorem 1.6.2 is not universal. That is, there is no constant A > 0such that |Φt(z)−z| ≤ A√t, for every z∈Dand every continuous semigroup {Φt}in D. Let us see why. Let { Φ t} be a continuous semigroup in D such that its Denjoy-Wolff point is zero. Then, by Theorem 1.3.3, its infinitesimal generator G:D→Cis of the form G(z) = −zp(z), with p : D→C+ (see Theorem 1.3.3). Then, by Theorem 1.6.5, there exists A > 0such that, |Φt(z)−z| ≤ A√t, for all z∈D. 29 1. Continuous semigroups of analytic functions We may assume that A is the best constant for this semigroup. Now, for b > 0, consider the semigroup { Ψ t} , where Ψ t ( z ) = Φ tb ( z ), z∈D , which is elliptic too and with the same Denjoy-Wolff point. In such case, we have that |Ψt(z)−z|=|Φtb(z)−z| ≤ A√t√b, where A1 ( b ) = A1 = A√b is the best constant for the semigroup { Ψ t} . However, the constant A1 is clearly not bounded, preventing the constant in the statement of Theorem 1.6.2 from being universal in the sense specified before. Remark 1.6.6.For the particular case of elliptic continuous semigroups { Φ t} with Denjoy-Wolff point zero and imposing the normalisation p (0) = 1 on the infinitesimal generator H = −zp ( z ), p : D→C+ , a slight modification in the proof of Theorem 1.6.2 allows us to provide a universal constant A > 0. Indeed, if p(0) = 1, Herglotz’s representation formula gives |p(z)| ≤ 2 1−|z|. Using this, considering p1:C+→C+,p1(w) = p(e−w), we obtain that |p1(w)|=|p(e−w)| ≤ 2 1−e−Re(w)≤4 Rew whenever 0 <Rew < 1. Moreover, there exists 0 < ε0< 1such that |p(z)|<4/ε0if |z|< e−1. Hence if ε<ε0and w∈Cε, we have that |p1(w)| ≤ 4 ε for all w∈Cε and ε < ε0 . Therefore, the constant K = 4 does the job and we can take A = A ( ρ, 4) as universal constant, where ρ is the constant from Corollary 1.5.8. Even though under some normalisations we are able to find a universal constant in the sense stated above, we do not know if the constant appearing is optimal. Question 1.6.7. Compute the best universal constant in the case the DenjoyWolff point is zero and p(0) = 1. 30 CHAPTER 2 Banach spaces of Dirichlet series Dirichlet series have been a long existing mathematical object. Euler was the first to work with them. Nonetheless, he just considered the later known as Riemann’s zeta function and computed its value at some real points. He also established his celebrated product formula. In 1775, Euler himself stated a problem on the existence of infinitely many primes in arithmetic progressions and proved it for some cases. Legendre, among other mathematicians, tried to prove the general statement of the theorem, but it was not until 1837 that Dirichlet provided a complete answer to the problem. He did so by introducing the now known as Dirichlet series and exploiting their properties, being the first to systematically use them in order to tackle a mathematical problem. This explains their popularity in the branch of Analytic Number Theory. However, their intrinsic interest for analysts working in Functional Analysis is quite recent. Indeed, in 1997, in their seminal paper [ 64 ], Hedenmalm, Linqvist, and Seip, following an intuition posed by Beurling in [ 41 ], introduced the first Banach spaces of Dirichlet series in order to tackle a problem in Harmonic Analysis. This chapter is split into two clear parts. Each of them is devoted to one of the notions appearing in the title of the chapter. In the first one, condensed in Section 2.1, we shall present the basic concepts regarding the definition and convergence of Dirichlet series. In the second part, some of the most relevant Banach spaces of Dirichlet series which will be object of study in forthcoming chapters will be presented. More specifically, in Section 2.2 we will recall the construction of the Hardy spaces version in the setting of Dirichlet series, namely, the Hp spaces, introduced in [ 64 ]. These spaces will be the ambient space in Chapter 3. In Section 2.5, we shall present the Bergman spaces of Dirichlet series Ap µ originally defined in 2015 by Bailleul and Lefèvre in [ 14 ]. In both Chapter 5and Chapter 6, we will study some problems related to these Bergman-type spaces. Because of the central role of composition operators in the next chapter, we will devote Section 2.3 to recall the characterisation of the bounded composition operators in Hp ,1 ≤p≤ ∞ . Throughout this chapter and in connection with the boundedness of composition operators on Banach spaces of Dirichlet series, we will be dealing with two classes of Dirichlet series symbols. Namely, G and G∞ . In Section 2.4, the closeness of the classes G and G∞ under the local uniform convergence in C+ is studied (see Theorem 2.4.2 and Corollary 2.4.4). Except Section 2.4 and Theorem 2.3.3, which are original results, the rest of the chapter is expository and well known to experts. Two basic references for 31 2. Banach spaces of Dirichlet series Similarly, by the Maximum Modulus Principle we have that ∥BP∥∞:= sup z∈D∞|BP(z)|= sup z∈T∞|BP(z)|= sup z∈Tr|BP(z)|. We claim that ∥BP∥∞=∥f∥∞.(2.4) Consider h:R→Tr, given by h(t) = (pit 1, . . . , pit r). Observe that P(it) = N X n=1 ann−it = N X n=1 an r Y j=1 (p−it j)αj(n)=BP(h(t)). The claim follows immediately from this observation and Kronecker’s Theorem. Indeed, as the map BP : Tr→C is continuous and, by Theorem 2.2.2, h ( R )is dense in Tr, we have that sup z∈Tr|BP(z)|= sup t∈R|BP(h(t))|= sup t∈R|P(it)|. 2.2.2 The space of bounded Dirichlet series The space of bounded Dirichlet series was introduced in 1997 by Hedenmalm, Linqvist, and Seip in [64]. Definition 2.2.3. The space H∞consists on the collection of Dirichlet series φ(s) = ∞ X n=1 ann−s convergent in C+such that ∥φ∥H∞= sup s∈C+|φ(s)|<∞. This is, the space H∞ is the Banach space of bounded convergent Dirichlet series in C+when endowed with the norm ∥·∥H∞(or simply ∥·∥∞). Remark 2.2.4.Bohr’s Theorem 2.1.6 allows us to define the space H∞ equivalently as the collection of those convergent Dirichlet series f having a bounded analytic extension to C+. Clearly, every Dirichlet polynomial is in H∞ . We shall now provide some examples of functions belonging to H∞ and others failing to belong to the space. Example 2.2.5. Let f ( z ) = P∞ n=1 bnzn be a function in H∞ ( D ). This is, the collection of bounded analytic functions in D . Then, we have that supz∈D|f ( z ) |<∞ . Given a prime number p , the function H ( s ) = f ( p−s ) belongs to H∞ , since it defines a convergent Dirichlet series in C+ and it is clearly bounded there. Notice that, in fact, the map ∆( f ) = F is nothing but applying the inverse Bohr lift, B−1 to the function f , for f a polynomial. It is straightforward that ∆is an isometric isomorphism between H∞(D)and H∞ Λ=H∞∩DΛ, 38 2.2. The Hardy spaces Hpof Dirichlet series where Λ = {pn : n∈N0} , for p a prime number. This is, H∞ Λ is the collection of bounded Dirichlet series depending only on the prime p . Notice also that this correspondence can be carried out for H∞ ( Dd ), d∈N and H∞ Λ , with Λ = {pn1 1···pnd d:nj∈N∪{0}}. Example 2.2.6. The function g ( s ) = 1 1+2−s is analytic in C+ and, in fact, can be written as a convergent Dirichlet series in such a half-plane. However, it fails to be bounded in C+. Therefore, gis not in the space H∞. Example 2.2.7. The function h ( s ) = e−s is clearly bounded in C+ . Nonetheless, it cannot be written as a convergent Dirichlet series there. Indeed, suppose that h could be represented by a convergent Dirichlet series P∞ n=1 ann−s , with {an} in C . Let N0 = inf{n : an = 0 }< + ∞ , if this were the case, then, thanks to Lemma 2.1.9, we would have that aN0= lim Re(s)→+∞Ns 0h(s) = (0if N0= 1,2, +∞otherwise. However, this contradicts the definition of N0. The following result is a Montel-type theorem for the space H∞ proven by Bayart in [15] and will be frequently used throughout the thesis. Theorem 2.2.8. ([ 15 , Lemma 11],[ 75 , Theorem 6.3.1]) Let {fn} be a bounded sequence in H∞ . Then, there exist both a subsequence {fnk} and a function f∈ H∞ , such that fnk converges uniformly to f on each half-plane Cε , for all ε > 0. 2.2.3 The Hardy spaces Hpof Dirichlet series for 1≤p < ∞ In the seminal paper [ 64 ], the first two Banach spaces of Dirichlet series were introduced. The first of them was H∞ , which we presented in Section 2.2.2. The second one was the Hilbert space H2 , which we are about to define. These spaces were considered as a tool to provide an answer to a long-standing question in Harmonic Analysis (see [ 64 ] or [ 75 , Section 6.1.1]). Some years later, Bayart showed the intrinsic interest of studying these spaces with the introduction in [ 15 ] of a new family of Hp -spaces of Dirichlet series. This definition, for the case p = 2, coincided of course with the one originally given in [ 64 ]. We shall carry out Bayart’s construction of these spaces in a slightly different way and show how it contains the Hilbert space H2as a particular case. As it was said before, the power of Bohr’s idea is that this identification between Dirichlet series and power series in the infinite polydisc D∞ is not just formal. Using Theorem 2.2.10, we can define a norm on the Dirichlet polynomials P by ‘lifting’ them to the infinite polydisc D∞ and computing the Lp ( T∞ )-norm of the extension to the distinguished boundary of D∞ (the infinite polytorus T∞ ) of this ‘lifted polynomial’. To do so, we need to use the next result from ergodic theory. Definition 2.2.9. Let x = ( x1, . . . , xd, . . . )be a sequence of real numbers Q - linearly independent. The Kronecker flow associated with the sequence x is the family of mappings Kt:T∞→T∞given by Kt(z)=(eix1tz1, . . . , eixdtzd, . . .), z = (z1, . . . , zd, . . .)∈T∞. 39 2. Banach spaces of Dirichlet series If e= (1,...,1, . . .),Kte:= (eix1t, . . . , eixdt, . . .). Theorem 2.2.10 (Birkhoff-Oxtoby, [75, Theorem 6.5.1] ).Let Kt be the Kronecker flow associated with the sequence ( x1, . . . , xd, . . . )of real numbers Q - linearly independent. Given a continuous function f:T∞→C, we have that lim T→+∞ 1 2TZT −T f(Kte)dt =ZT∞ f(z)dm∞(z).(2.5) Now, consider the sequence of Q -linearly independent real numbers x = ( log p1,log p2, . . . ), where, we recall, {pj}j is the sequence of prime numbers. Then, Kte = ( pit 1, . . . , pit d, . . . ). Let P be a Dirichlet polynomial and set f ( z ) = |BP ( z ) |p , which is clearly a continuous function in T∞ . Notice that with the choice we did of the sequence x, BP(Kte) = BP(pit 1, . . . , pit r, . . .) = X n≥1 n=pα1 1···pαr r an(p−it 1)α1. . . (p−it r)αr = ∞ X n=1 ann−it =P(it). Therefore, f(Kte) = |P(it)|pand we can define the following norm on P, ∥P∥Hp:= lim T→∞ 1 2TZT −T|P(it)|pdt!1 p .(2.6) The fact that ∥·∥Hpis a norm is a consequence of Theorem 2.2.10 since lim T→∞ 1 2TZT −T|P(it)|pdt!1 p =ZT∞|BP(z)|pdm∞(z) = ∥BP∥Lp(T∞). Theorem 2.2.11. ([ 75 , Thereom 6.5.2]) For each P∈ P and 1 ≤p < ∞ , the functional ∥P∥Hp:= lim T→∞ 1 2TZT −T|P(it)|pdt!1 p (2.7) defines a norm in P. As it was mentioned previously, we shall use a different approach to address the construction of the Hp -spaces originally carried out by Bayart in [ 15 ] (see also [ 75 , p. 166]). The idea behind Bayart’s construction is defining a norm on the Dirichlet polynomials P by computing the Lp ( T∞ )-norm of the restriction to the distinguished boundary of D∞ , this is, T∞ , of the Bohr-lift of the Dirichlet polynomials. Once this is done, the spaces Hp are defined as the completion of the Dirichlet polynomials with respecto to this norm. To know where the resulting Dirichlet series converge pointwise, it is required to prove beforehand the boundedness of the pointwise evaluations functionals, so that the norm convergence implies local uniform convergence in a certain half-plane. In order to avoid this, but still being able to deduce from the construction of the spaces 40 2.2. The Hardy spaces Hpof Dirichlet series an initial convergence half-plane, namely C1 , we shall start from the better known spaces Hp ( T∞ ),1 ≤p < ∞ . This is, the completion in the Lp ( T∞ )-norm of the trigonometric polynomials with vanishing Fourier coefficients outside the narrow cone N(∞) 0 , the sequences {nj}j of non-negative integers with only finitely many non-zero terms. More precisely, Hp(T∞) = {f∈Lp(T∞) : b f(α)=0if α∈ N(∞) 0}, where b f ( α )stands for the Fourier coefficient associated with the term zα = zα1 1···zαr r. The good thing about the space Lp ( T∞ )is the existence of approximations of the identity. Then, the inverse Bohr-lift will allow us to take the completion of trigonometric polynomials in the Lp ( T∞ )-norm to the Dirichlet polynomials in order to obtain a closure with the expected properties (convergent Dirichlet series in C1). For this purpose, we define, for 1≤p < ∞, Hp=B−1f:f∈Hp(T∞). Let us check that that this definition gives convergent Dirichlet series in C1. We will use an approximation of the identity formed by trigonometric polynomials in T∞ . In fact, there exists a sequence of trigonometric polynomials in T∞with the following properties: i)∥Hn∥L1(T∞)≤1,for all n;ii)c Hn(α)→1,for all α∈Z(∞). (2.8) Then, for every f∈Lp(T∞),1≤p < ∞, we have lim n→∞ ∥Hn∗f−f∥Lp(T∞)= 0.(2.9) If f∈Hp ( T∞ )and f is a trigonometric polynomial, by the definition of the Bohr lift from Subsection 2.2.1, we first observe that B−1f(s) = ∞ X n=1 ann−s, where an = b f ( k1, . . . , kr, 0 , . . . ), n = pk1 1, . . . , pkr r . Taking this into account, for each n , the convolution Hn∗f defines a trigonometric polynomial in Hp ( T∞ ). Hence, by the previous observation, we have that Hn∗f = BPn , where for each n,Pn(s) = Pk≥1an kk−sis a Dirichlet polynomial. Not only this, but •the sequence {BPn}nis a Cauchy sequence in the Lp(T∞)-norm; •limn→∞ an k = \ B−1f ( k ), for all k∈N . This is a consequence of both the boundedness of the functional f7→ b f(α)in Hp(T∞)and (2.9). This implies that the limit sequence of coefficients, say {ak}k = {\ B−1f ( k ) }k , of the Dirichlet series B−1f is bounded. Hence, the Dirichlet series in Hp converge, a priori, in C1 . We thus obtain a family of Banach spaces of analytic functions a priori defined in C1. 41 2. Banach spaces of Dirichlet series Definition 2.2.12. The spaces Hp consist on those analytic functions in C1 representable by a Dirichlet series, which is the pointwise limit in C1 of a Cauchy sequence {Pj}in the norm ∥·∥Hpof Dirichlet polynomials Pj= ∞ X n=1 a(j) nn−s.(2.10) Moreover, given f∈ Hp, we define ∥f∥Hp:= lim j→∞ ∥Pj∥Hp,(2.11) where {Pj}jis the sequence defined in (2.10). Let us remark that a function f in Hp completely determines its Bohr lift Bf . Indeed, consider f∈ Hp . Then, there exists a Cauchy sequence of Dirichlet polynomials {Pj} converging to f in the Hp -norm and pointwise to f in C1 . Then, the sequence {BPj} is a Cauchy sequence in Hp ( T∞ ). In particular, this implies the existence of a function F in Hp ( T∞ )such that BPj→F in Hp ( T∞ ) and, moreover, d BPj(α)→b F(α), for all α∈Z(∞). Knowing this, we have that lim j→∞ Pj(s) = ∞ X n=1 ann−s, pointwise in C1 , where an = b F ( α ( n )). Now, since, Pj→f pointwise in C1 , then f(s) = P∞ n=1 ann−sand, by the uniqueness of the limit, F=Bf, as desired. This observation implies that the definition of the Hp -spaces from Definition 2.2.12 does not depend on the sequence {Pj} of Dirichlet polynomials chosen. More precisely, given two sequences {Pj} and {Qj} satisfying the conditions from Definition 2.2.12, we have that ∥Pj−Qj∥Hp→0, as j→ ∞. Observe that in the construction of the spaces Hp carried out so far, the case H∞ has been excluded. However, the space of bounded Dirichlet series can be embedded in such construction: Proposition 2.2.13. H∞={B−1f:f∈H∞(T∞)}. In fact, H∞consists on those H2functions fsuch that Bf∈H∞(T∞). Proof. Let f∈H∞ ( T∞ ). We want to show that B−1f belongs to H∞ . First, let {Hn} be a sequence of trigonometric polynomials in T∞ satisfying i )and ii ) from (2.8). Then, ∥Hn∗f∥H∞(T∞)≤ ∥f∥H∞(T∞)≤C. Recalling that, for each n∈N , Hn∗f is a trigonometric polynomial, by (2.4) , we have that ∥B−1(Hn∗f)∥H∞=∥Hn∗f∥H∞(T∞)≤C. Set Pn = B−1 ( Hn∗f ). Then, the sequence {Pn} is a bounded sequence in H∞ . Applying Theorem 2.2.8 to {Pn} , we know the existence of both a function g 42 2.2. The Hardy spaces Hpof Dirichlet series in H∞ and a subsequence {Pnk} such that Pnk→g uniformly in Cε , ε > 0, as k→ ∞ . Now, since H∞ ( T∞ ) ⊂H2 ( T∞ ), we have that Hn∗f→f in H2 ( T∞ ), as n→ ∞. Hence, repeating the argument in page 41, we have that lim n→∞ B−1(f∗Hn)(s) = lim n→∞ Pn(s) = B−1f(s), s ∈C1. The uniqueness of the limit gives that g=B−1f, as desired. Suppose now that f belongs to H∞ . Then, for every ε > 0, B ( fε )is in H∞(T∞). Indeed, ∥B(fε)∥H∞(T∞)=∥fε∥H∞≤ ∥f∥H∞. We now use the fact that H∞⊂ H2 (see, for instance, [ 75 , Theorem 6.2.5]). Then, fε→f in H2 as ε→ 0 + . Consequently, B ( fε ) → Bf in H2 ( T∞ )as ε→ 0 + . In particular, B ( fε ) → Bf a.e. in T∞ . Now, since B ( fε ) ∈H∞ ( T∞ ), for all ε > 0, we conclude that |Bf| ≤ C a.e. in T∞ , giving the desired conclusion. ■ This result allows us to define the Hp spaces in a same fashion for every p∈ [1 ,∞ ]. We then obtain a family of Banach spaces of Dirichlet series with the expected inclusion relations between the spaces. Proposition 2.2.14. Let 1 ≤p≤q≤ ∞ . Then, Hq⊂ Hp and given f∈ Hq , ∥f∥Hp≤ ∥f∥Hq. For p∈ [1 , + ∞ ], we will write Hp ∞ to denote the subspace of Hp whose elements have vanishing first coefficient i.e. those functions f in Hp such that limRe(s)→+∞f(s)=0. Let us now address the question of the maximal convergence half-plane. To do so, we will do a detour through the space H2 . Let P ( s ) = P∞ n=1 ann−s be a Dirichlet polynomial. A direct computation shows that ∥P∥2 H2= lim T→∞ 1 2TZT −T|P(it)|2dt = ∞ X n=1 |an|2. This identity is in fact a consequence of the more general result known as Carlson’s identity (see [ 64 , Lemma 3.2] or [ 75 , Theorem 6.2.5]). Then, we also could have defined the space H2 ([ 64 ]) as the collection of formal convergent Dirichlet series with square summable coefficients. This is, H2=  f(s) = ∞ X n=1 ann−s:∥f∥H2= ∞ X n=1 |an|2!1 2 <∞  .(2.12) This definition coincides with the one given in Definition 2.2.12. However, notice that the definition of H2 from (2.12) gives, by Cauchy-Schwarz inequality, that σa ( f ) ≤ 1 / 2for every f∈ H2 . Moreover, the bound 1 / 2cannot be improved, meaning that for every r < 1 / 2, there exists a Dirichlet series f in H2 failing to converge in Cr . Nonetheless, in Hp , the functions are defined, a priori, by a convergent Dirichlet series in C1 . The fact that the Hp is a space of analytic functions in C1/2 is a consequence of the boundedness of the pointwise evaluations functionals in C1/2 . We recall that for a Banach space X of analytic 43 2. Banach spaces of Dirichlet series functions in a domain Ω ⊂C and z0∈ Ω, the evaluation functional at the point z0,δz0, acting on f∈Xis defined as δz0(f) = f(z0). Theorem 2.2.15. ([ 15 , Theorem 3]) Let 1 ≤p < ∞ . Consider f∈ Hp . Then, the Dirichlet series defining f converges in the half-plane C1/2 . Moreover, for s=σ+it,σ > 1/2, we have that |f(s)| ≤ ζ(2σ)1 p∥f∥Hp, and ∥δs∥(Hp)∗=ζ(2σ)1 p, where ζis the Riemann zeta function. Observe that this theorem gives that σu ( f ) = σc ( f ) ≤ 1 / 2, f∈ Hp . Then, the Hpspaces are Banach spaces of analytic functions in the half-plane C1/2. It is worth mentioning that, in fact, we have that σa ( f ) ≤ 1 / 2for every f∈ Hp , p∈ [1 ,∞ ). This is a consequence of a more general result due to Helson. Actually, we could have deduced the convergence in C1/2 without passing through the functionals δs . However, as we will need them in the next chapter, we have found convenient to introduce them at this point of the exposition. Nonetheless, let us state Helson’s result and see how it concerns the convergence of functions in the spaces Hp. Theorem 2.2.16. ([ 65 , p.89], [ 75 , Theorem 6.5.9]) Let f ( s ) = P∞ n=1 bnn−s∈ H1 . Then, ∞ X n=1 |bn|2 d(n)!1 2 ≤ ∥f∥H1,(2.13) where d(n)denotes the number of divisors of n. In particular, σa(f)≤1/2. Let us see why (2.13) gives that σa ( f ) ≤ 1 / 2. We recall the following elemental result from Number Theory. Lemma 2.2.17. ([ 7 , Theorem 13.12]) Let n be a nonnegative integer and let d ( n )denote the number of divisors of n . Then, for every ε > 0, there exists a positive constant C=C(ε)such that d(n)≤Cnε. Let s = σ + it , σ > 1 / 2. We choose ε > 0such that 2 σ−ε > 1. Then, applying Cauchy-Schwarz’s inequality first and then using both Theorem 2.2.16 and Lemma 2.2.17, ∞ X n=1 |bn| nσ≤ ∞ X n=1 |bn|2 d(n)!1 2 ∞ X n=1 n−2σd(n)!1 2 ≤C∥f∥H1 ∞ X n=1 1 n2σ−ϵ!1 2 <∞. The following uniform bound of the norm of the evaluations of the derivatives in H2will be used in Chapter 3. For k∈N∪ { 0 } and f an analytic function, f(k) stands for the k -th derivative of f. For k= 0,f(k)=f. Lemma 2.2.18. ([ 48 , Lemma 2.4]) Let k∈N∪{ 0 } and s∈C1/2 . The functional δk s:H2→Cgiven by φ7→ φ(k)(s) is bounded. In fact, given σ > 0, it is uniformly bounded for s∈C1/2+σ. 44 2.2. The Hardy spaces Hpof Dirichlet series Proof. Writing φ(s) = P∞ n=1 ann−s, notice that φ(k)(s) = ∞ X n=1 (−log n)kann−s, s ∈C1/2. Fix σ > 0. For s∈C1/2+σ, Cauchy-Schwarz’s inequality gives |φ(k)(s)| ≤ ∞ X n=1 (log n)k|an| nRe(s)≤ ∞ X n=1 (log n)2k n1+2σ!1/2 ∞ X n=1 |an|2!1/2 <∞. Then, |φ(k)(s)| ≤ C(σ)∥φ∥H2,s∈C1/2+σ.■ Remark 2.2.19.The previous lemma is still valid for the Hp -spaces, 1 ≤p < ∞ . For p≥ 2, the previous lemma is clearly true in Hp . For p∈ [1 , 2), Cauchy’s inequality and Theorem 2.2.15 give the result. Let us sketch the proof. Let f∈ Hp and σ > 0. Choose s∈C1/2+σ and let r = Re ( s ) − 1 / 2. For k∈N0, by Cauchy’s inequality, we have that |f(k)(s)| ≤ (r/2)−ksup z∈D(s,r/2) |f(z)| ≤ (σ/2)−ksup Re(z)= 1 2(Re(s)+1/2) |f(z)| ≤(σ/2)−ksup Re(z)= 1 2(σ+1) |f(z)| ≤(σ/2)−kζ(σ+ 1)1 p∥f∥Hp, where in the last inequality we have used Theorem 2.2.15. Naturally, the later proof holds for every p∈ [1 ,∞ ). Nonetheless, the proof given for Lemma 2.2.18 is more elementary and does not involve deeper results like Theorem 2.2.15. We conclude this section with a remarkable result obtained by Aleman, Olsen, and Saksman and which will be needed in Chapter 6. Theorem 2.2.20. ([ 4 , Corollary 4]) Let en = n−s , n∈N and 1 <p<∞ . Then, {en}is a Schauder basis for Hp. 2.2.4 Vertical and horizontal translations Let f be a Dirichlet series absolutely convergent in a certain half-plane Cθ , θ∈R. We define its vertical translations as the family of functions given by fτ(s) = f(s+iτ), τ ∈R, s ∈Cθ. Observe that given a sequence of reals {τk}k , the associated translations {fτk}k are uniformly bounded in each compact subset K of Cθ . Hence, since σb ( f ) ≤σa ( f ), we can apply Montel’s theorem which guarantees the existence of a subsequence, {τkj}j such that the functions {fτkj}j converge uniformly on compacts of Cθ to some function e f . We then say that e f is a vertical limit of f . In fact, it is possible to characterise all the vertical limits of the functions f . 45 2. Banach spaces of Dirichlet series Lemma 2.2.21. ([ 16 , Lemma 2.1]) Let f ( s ) = P∞ n=1 ann−s be a Dirichlet series absolutely convergent in some half-plane Cθ. Then, the vertical limits of fare given by the functions fχ(s) = ∞ X n=1 anχ(n)n−s,(2.14) where χ∈T∞. Notice that since we are multiplying the general term by a bounded function, the absolute convergence abscissa of fχ coincides with the one for f . This, in particular, implies that fχ∈ D. Moreover, we have that, sup s∈Cθ|fχ(s)| ≤ sup s∈Cθ|f(s)|. In particular, if f belongs to H∞ , so does fχ . The latter inequality is in fact an equality. Indeed, the role of the vertical limits is symmetric, since f = ( fχ ) χ−1 . We also have an equality when passing to the Bohr-lift since taking vertical limits acts as a rotation in T∞. Indeed, B(fχ)(z) = Bf(χz),z∈T∞. The vertical limits may improve the pointwise convergence of the Hp - functions. The following result will be of great importance in Chapter 6in order to obtain Littlewood-Paley type identities for the Hp-norm. Theorem 2.2.22. ([ 15 , Theorem 6]) Let 1 ≤p < ∞ and f∈ Hp . Then, for almost every χ∈T∞,fχis a Dirichlet series convergent in C+. We now introduce the horizontal translations operator which will repeatedly appear throughout the thesis. We define the operator Tε acting on a convergent Dirichlet series f(s) = P∞ n=1 ann−s, as Tεf(s) = f(s+ε) = ∞ X n=1 ann−εn−s. At some points of the exposition, specially in Chapter 5and Chapter 6, in order to ease the reading, we shall write fε , instead of Tεf , meaning that we are considering the horizontal translation of the Dirichlet series f , f ( s + ε ), instead of the operator Tεacting on the function f. It is noteworthy that the Hp spaces remain stable under horizontal translations. This is, Tε : Hp→ Hp boundedly. This result was first established in [ 64 , Lemma 3.3] for the case p = 2 as a corollary of the Carlson’s identity ([ 64 , Lemma 3.2]). We state some properties of the translated functions fε straightaway. Theorem 2.2.23. ([50, Proposition 11.20]) Let 1≤p < ∞and f∈ Hp. Then, a) ∥f∥Hp= supε>0∥fε∥Hp. b) ε7→ ∥fε∥Hpis a decreasing on [0,∞). c) limε→0+∥f−fε∥Hp= 0. A question which arises when considering horizontal translations is that of whether for f∈ Hp we could have that fε∈ Hq , with q > p . This question was answered affirmatively by Bayart in [ 16 ]. In fact, he proved a more general result. We recall that a sequence {λn}n of nonnegative integers is said to be completely multiplicative if λmn =λmλnfor all m, n ≥1. 46 2.3. Gordon-Hedenmalm class: composition operators on Hp-spaces Theorem 2.2.24. ([ 16 , Corollary 3.3]) Let 1 ≤p≤q < ∞ . Consider a completely multiplicative sequence {λn} and f ( s ) = P∞ n=1 ann−s∈ Hp . If λn≤pp/q for nbig enough, then the multiplication operator Mλn(f) = ∞ X n=1 anλnn−s is bounded from Hpinto Hq. Clearly, the sequence λn = n−ε , ε > 0is under the hypothesis of Theorem 2.2.24. Hence, we have this immediate consequence. Corollary 2.2.25. Let 1 ≤p≤q < ∞ and ε > 0. Then, the translation operator Tε:Hp→ Hqis bounded. Remark 2.2.26.Bayart also proved that whenever ε > 0is large enough, the operator Tεis a contraction ([15, Theorem 10]). 2.3 Gordon-Hedenmalm class: composition operators on Hp-spaces A natural question which emerges when studying Banach spaces theory is that of the boundedness of some remarkable linear operators. Among those, one which usually attires special attention is the composition operator. Given a Banach space X of analytic functions in a domain Ω ⊂C , the composition operator of symbol Φis the operator CΦ , defined by CΦf = f◦ Φ, f∈X . The characterisation of bounded composition operators CΦ acting on some Banach space X aims to provide a necessary and sufficient condition on Φensuring boundedness. In the context of the Hp spaces, the firsts to study composition operators were Gordon and Hedenmalm. In their seminal work [ 61 ], Gordon and Hedenmalm focused their attention on composition operators in the Hilbert space H2 . More precisely, given an analytic function Φ : C1/2→C1/2 , they wondered about the minimal requirements for a given analytic function symbol to define a bounded composition operator in H2 . Reciprocally, once we have the boundedness of such an operator, what properties does Φsatisfy? To answer this matter, in [ 61 ] the authors introduced the now so-called Gordon-Hedenmalm class, denoted by G , even though they did not use this name in their original paper. Definition 2.3.1. Given an analytic function Φ : C+→C+ , we say that Φ belongs to the Gordon-Hedenmalm class Gif: 1) There exists cΦ∈N∪{0}and φa Dirichlet series such that Φ(s) = cΦs+φ(s), s ∈C+.(2.15) 2) If cΦ= 0, then Φ(C+)⊂C1/2. The value cΦis known as the characteristic of the function Φ. 47 2. Banach spaces of Dirichlet series With the definition (2.19) we obtain a Hilbert space of Dirichlet series in the half-plane C1/2 . This is due to the fact that for every f∈ A2 µ , σa ( f ) ≤ 1 / 2. Indeed, consider f ( s ) = Pn≥1ann−s in A2 µ . Let ε > 0and s∈C1/2 . By Cauchy-Schwarz inequality and (2.18), we have that ∞ X n=1 |an|n−Re(s)≤ ∞ X n=1 |an|2wn!1 2 ∞ X n=1 1 n2 Re(s)wn!1 2 ≤ ∥f∥A2 µ ∞ X n=1 c n2 Re(s)+ε!1 2 <∞. Not only we have that the elements in A2 µ are defined in C1/2 , but, in fact, this domain is maximal, meaning that not every element in the space can be defined in a strictly larger half-plane. To see this, for each ε > 0, it suffices to consider the function g(s) = ∞ X n=1 1 n1 2+s+ε· For every ε > 0, g belongs to every A2 µ thanks, once more, to (2.18) . However, it has a pole at the point s= 1/2−ε. 2.5.2 The Bergman spaces Ap µ Now, the question is how to pass from the hilbertian case to the range p∈ [1 ,∞ ). In order to do so, we are going to define the following norm on the Dirichlet polynomials. Definition 2.5.3. Let µ be an admissible measure on (0 , + ∞ )and P be a Dirichlet polynomial. We define the norm ∥P∥Ap µ:= Z+∞ 0∥Pσ∥p Hpdµ(σ)1 p ,(2.21) where Pσ(s) = P(s+σ). Observe that, with this definition, we can see the weights from Definition 2.5.1 from a different angle: wn=Z+∞ 0 n−2σdµ(σ) = ∥en∥2 A2 µ, where en is the monomial n−s , n∈N . Another immediate observation is that Hölder’s inequality gives, for Pa Dirichlet polynomial and 1≤p≤q < ∞, ∥P∥Ap µ≤ ∥P∥Aq µ. The spaces Ap µ are defined as the completion of the Dirichlet polynomials with respect to the norm (2.21) . To give a precise sense to this completion procedure and, more importantly, in order to see what elements lie in the closure, we will sketch a similar argument to the original one carried out in [ 14 , Section 54 2.5. Bergman spaces of Dirichlet series 2]. Before continuing, let us remark that the definition for the space A2 µ from (2.19) and this new definition in terms of the norm from (2.21) coincide (see [14]). Also notice that the spaces Ap µ will satisfy the expected inclusion relations, this is, Aq µ⊂ Ap µ,1≤p≤q < ∞. We shall also consider an important subspace of Ap µ , namely, the space Ap µ,∞ , consisting on those elements f ( s ) = Pn≥1ann−s in Ap µ with vanishing first coefficient i.e. a1 = limRe s→+∞f ( s ) = 0. Because of this, we may sometimes refer to this space as the subspace of Ap µfunctions vanishing at infinity. Being the case p = 2 a particular example of this construction, we somehow expect the Ap µ spaces to give a family of Banach spaces of analytic functions defined in C1/2 . To see that this is in fact the case, we first need to establish the boundedness of the pointwise evaluations at a point s in the half-plane C1/2 . Theorem 2.5.4. ([ 14 , Theorem 1]) Let p≥ 1and µ be an admissible probability measure. Then, the point evaluation at any s∈C1/2 is bounded on Ap µ (respectively Ap µ,∞). In Chapter 5we will address the problem of estimating the norm of these evaluations when µ = µα , α > − 1. This estimate will be of great interest in Chapter 6in connection with the boundedness of the Volterra operator Tg when acting on the spaces Ap α . Because of this, we will leave the details on these functionals for Chapter 5. The next Lemma is used in Theorem 2.5.7 in order to define the Ap µ -norm for every function in the space. We recall that fε=Tεf,ε > 0. Lemma 2.5.5. Let 1 ≤p < ∞ , ε > 0and µ be an admissible measure. Then, the operator Tε:P ∩Ap µ→ Hp, f 7→ f(·+ε) extends to a bounded operator on Ap µ. Proof. Let f be a Dirichlet polynomial. By the definition of the Ap µ -norm on the Dirichlet polynomials and using Theorem 2.2.23 b) twice, we have that ∥f∥p Ap µ=Z+∞ 0∥fσ∥p Hpdµ(σ)≥Z+∞ 0∥fσ+ε/2∥p Hpdµ(σ) ≥Zε/2 0∥fσ+ε/2∥p Hpdµ(σ) ≥µ([0, ε/2)]∥fε∥p Hp. Since 0 ∈supp ( µ ), we have µ ((0 , ε/ 2)) > 0. The conclusion follows by a density argument. ■ Actually, with some extra work, we can prove a better bound (see Lemma 6.2.4): Tεmaps boundedly Ap µinto every Hq,1≤p, q < ∞. Let f∈ Ap µ . Let us see that f is a convergent Dirichlet series on C1/2 . By the inclusion relations between the spaces Ap µ , it suffices to prove the statement for the case p= 1. 55 2. Banach spaces of Dirichlet series Let f belong to A1 µ and ε > 0. Then, there exists a sequence {Pj} of Dirichlet polynomials, Pj ( s ) = Pn≥1aj nn−s , converging to f in the A1 µ -norm. In particular, by Theorem 2.5.4, for each s∈C1/2 , Pj ( s )is a Cauchy sequence in C . Hence, there exists g such that Pj ( s ) →g ( s )as j→ ∞ for all s∈C1/2 . By Lemma 2.5.5, the function fε/2 is in H1 . Since the coefficients are continuous in P ∩H1 , for each n∈N the sequence {bj n}j , bj n = aj nn−ε/2 is also continuous. Hence, for all n∈N , the sequence {aj n}j is continuous too. Consequently, there exists a sequence {an} such that limj→∞ aj n = an for all n∈N . Now, for each j, the sequence {bj n}jis bounded for all n. Then, by Weierstrass M-test, lim j→∞ Pj(s) = ∞ X n=1 ann−s=f(s), s ∈C1+ε, and f = g in C1+ε . By the uniqueness of holomorphic continuation, we have f = g in C1/2 . The sequence {Pj} is bounded in H∞ ( C1/2+ε ). Since Pj→f pointwise in C1/2 , we conclude that f∈H∞ ( C1/2+ε ), ε > 0. Moreover, f is in D , because it is defined in C1+ε by a convergent Dirichlet series. Therefore, we can apply Bohr’s Theorem (see Theorem 2.1.6) to conclude that f∈ H∞ ( C1/2+ε ) for all ε > 0. Since ε > 0is arbitrary, the series defining f converges pointwise in C1/2. In particular, σu(f)≤0, as desired. Theorem 2.5.6. ([14, Theorem 5]) For every f∈ Ap µ, we have that f∈ D and σu(f)≤1/2. The norm (2.21) can be written explicitly when f is an Hp function. For a general Ap µ -function, using the fact that Tε ( Ap µ ) ⊂ Hp from Lemma 2.5.5, we can compute its norm as follows: Theorem 2.5.7. ([ 14 , Theorem 6]) Let p≥ 1and µ be an admissible measure. Then, i) Hp⊂ Ap µand ∥f∥Ap µ≤ ∥f∥Hpfor every f∈ Hp. ii) For every f∈ Hp,∥f∥Ap µ=R+∞ 0∥fσ∥p Hpdµ(σ)1 p. iii) For every f∈ Ap µ ∥f∥Ap µ= lim ε→0+∥fε∥Ap µ. As in the case p = 2, let us mention that the consideration of the Dirac mass at 0, again gives the Hp-spaces as a limiting case. In Chapter 6having Littlewood-Paley identities for the Ap µ norm (see [ 14 , Theorem 7]) will be crucial. Since these identities will consider Dirichlet series defined in C+ , the following theorem is of capital importance. Its proof is essentially the same as in the Hpcase. Theorem 2.5.8. ([ 14 , Proposition 2]) Let 1 ≤p < ∞ and f∈ Ap µ . Then, for almost every χ∈T∞,fχis a Dirichlet series convergent in C+. Dirichlet monomials {en} are also a Schauder basis for Ap µ for 1 < p < ∞ . This is in fact an immediate consequence of Theorem 2.2.20. 56 2.5. Bergman spaces of Dirichlet series Theorem 2.5.9. ([ 14 , p. 26]) Let en = n−s , n∈N and 1 <p<∞ . Then, {en} is a Schauder basis for Ap µ. 57 CHAPTER 3 Semigroups of composition operators on Hardy spaces of Dirichlet series Previously in our exposition, we have introduced some concepts which shall come into play in the upcoming pages. In the first chapter, the notion of continuous semigroups in D and in C+ was broadedly treated. In this chapter, we shall introduce the notion of strongly continuous semigroups of operators in Banach spaces. Essentially, these semigroups are families of operators satisfying the standard algebraic relations as in the case of semigroups of analytic functions, which also happen to converge to the identity in the strong operator topology. The semigroups of analytic functions { Φ t} and the semigroups of operators {Tt} can be linked and their properties are susceptible of being studied simultaneously by considering the composition operators Tt = CΦt , already introduced in Chapter 2. In 1978, Berkson and Porta [ 22 ] were the firsts to use this approach in the setting of Banach spaces of analytic functions. More precisely, they proved that a semigroup of composition operators {CΦt} is strongly continuous in every Hardy space of the unit disc, Hp ( D ), for 1 ≤p < ∞ , if and only if the family of functions { Φ t} is a continuous semigroup of holomorphic functions in the unit disc. This research was later extended to other spaces, having an analogue characterisation in other classical spaces like Bergman spaces, but rather different in others cases like the disc algebra, the Bloch space or BMOA, among others. See [6], [8], [24], [25], [46], [56], and references therein. In this chapter, we will establish an analogue of Berkson and Porta’s result in the context of Hardy spaces of Dirichlet series. More precisely, in Section 3.2 we will establish a one-to-one correspondence between continuous semigroups of analytic functions in the class G and strongly continuous semigroups of composition operators in the Hp spaces, p∈ [1 ,∞ ). For the case p = ∞ , we will show in Section 3.4 that the only strongly continuous semigroup of composition operators in H∞ is the trivial one, this is, Tt = Id , for all t≥ 0. However, it is not clear beforehand that there exist non-trivial continuous semigroups in the class G . In Section 3.3, we give a description of the infinitesimal generators of these semigroups, which provides a wide range of examples of continuous semigroups in the class G . Section 3.5 is devoted to the characterisation of the Koenigs functions of the continuous semigroups in the class G , as well as the study of more dynamics-related properties. Some of the results established in 59 3. Semigroups of composition operators on Hardy spaces of Dirichlet series this section will be needed in Chapter 4. Eventually, in section 3.1 we shall prove some properties related to Dirichlet series as well as some key lemmas in connection with the properties of the elements of continuous semigroups in the class G. 3.1 Preliminary results We present the basic results that will be needed in further sections. The first of them is probably well-known for specialists, but since we could not find a reference we include its proof. Theorem 3.1.1. Let φ ( s ) = P∞ n=1 ann−s be a Dirichlet series convergent in Cη , η∈R. Then, φis not one-to-one in Cη. Proof. Take N = min{n≥ 2 : an = 0 } . We may assume that a1 = 0 and aN= 1. Write φ(s) = N−s+h(s)and γ(s) = N−sfor all s∈Cη. Take ε < ( N− 1) /N2 . Since limRe s→+∞h ( s ) Ns = 0 (see Lemma 2.1.9), there is s0> η such that |h(s)|< N−Re sεwhenever Re s > s0. Take x > s0 + 1 and m∈Z . Write g ( s ) := N−s−N−x and f ( s ) := φ(s)−N−x, for s∈Cη. Consider the segments Γa,m = [a−iπ/ ln(N), a +iπ/ ln(N)] + 2mπi/ ln(N), ∆a,m = [a−iπ/ ln(N), a + 2 −iπ/ ln(N)] + 2mπi/ ln(N), for every a∈R . Notice that γ (Γ a,m )is the circle centered at 0and with radius N−a , that is C (0 , N−a ), and γ (∆ a,m )is the segment that joins the points −N−awith −N−(a+2). Now, take the rectangle Γm= Γx−1,m ∪Γx+1,m ∪∆x−1,m ∪∆x−1,m+1. On the one hand, notice that Γ m is contained in the half-plane Cs0 . Thus, for any s∈Γm, it holds |g(s)−f(s)|=|h(s)|< N−Re sε≤N−(x−1)ε. (3.1) On the other hand, observe that γ(Γm) = γ(Γ0) = C(0, N−(x−1))∪C(0, N−(x+1))∪[−N−(x−1),−N−(x+1)]. Thus, for s∈Γm, we have |g(s)|=|γ(s)−N−x|=|N−s−N−x| ≥min{|N−(x−1) −N−x|,|N−(x+1) −N−x|} ≥ N−x−1(N−1).(3.2) Since ε < (N−1)/N2, we deduce that for all s∈Γm, (3.1) and (3.2) imply |g(s)−f(s)|<|g(s)|. The point x + i 2 mπ/ ln ( N )is the center of Γ m and g ( x + i 2 mπ/ ln ( N )) = 0 (in fact, it is the unique zero of g in the interior of such rectangle). Now, Rouché’s Theorem implies that the equation φ ( s ) = N−x has a solution in the interior of each Γmwhat clearly shows that φis not univalent. ■ 60 3.1. Preliminary results In Remark 2.1.8, the following theorem was mentioned but not explicitly stated. We state it at this point since it will be repeatedly used throughout this chapter. The reader may find a proof of it in [76] or [75, Theorem 8.4.1]. Theorem 3.1.2. Let σ, ν ∈R . Consider φ : Cσ→Cν analytic such that it can be written as a Dirichlet series in a certain half-plane. Then, σu ( φ ) ≤σ . In particular, φis bounded in Cσ+εfor all ε > 0. In connection with this theorem, it is also useful to introduce the following family of Banach spaces of Dirichlet series, namely, H∞ ( Cε ), ε > 0. These spaces consist on those Dirichlet series φconverging in Cεsuch that ∥φ∥H∞(Cε):= sup s∈Cε|φ(s)|<∞. Observe that if we take ε = 0 we recover the space H∞ from Chapter 2. As it was the case for this space, the spaces H∞ ( Cε )are Banach algebras too. Once more, thanks to Bohr’s Theorem 2.1.6, these spaces coincide with the ones consisting on those somewhere convergent Dirichlet series having a bounded analytic extension to the half-plane Cε. The next results describe some properties of the elements in the class G . More specifically, the first two are generic properties of the functions in this class, whereas the other two results concern the properties of the iterates of continuous semigroups in the class G. Lemma 3.1.3. Let Φbelong to the class G and be such that cΦ∈N . Then, σu(φ)≤0, where, as usual, Φ(s) = cΦs+φ(s),s∈C+. Proof. By Lemma 2.1.9, it holds cΦ= lim Re s→+∞ Φ(s) s. Thus, by Julia-Wolff-Carathéodory’s Lemma(see [ 35 , Theorem 1.7.8]), Re Φ( s ) ≥ cΦRes for all s∈C+ . Therefore, φ sends C+ into C+ . In addition, by Theorem 3.1.2, we have that σu(φ)≤0.■ The classes G and G∞ are stable under composition and the characteristic of the composition is the product of the characteristics. This fact is relevant when dealing with semigroups of functions in these classes. Proposition 3.1.4. Let Φ , Ψ ∈ G∞ (resp. Φ , Ψ ∈ G ). Then, Φ ◦ Ψ ∈ G∞ (resp. Φ◦Ψ∈ G). Moreover, cΦ◦Ψ=cΦcΨ. Proof. Given Φ , Ψ ∈ G∞ and f∈ H∞ , Theorem 2.3.8 guarantees that f◦ Φ ∈ H∞ . Now, once more, f◦ (Φ ◦ Ψ) = ( f◦ Φ) ◦ Ψ ∈ H∞ . Then, Φ ◦ Ψis a symbol of a bounded composition operator from H∞ into H∞ . Therefore, again by Theorem 2.3.8, necessarily, Φ ◦ Ψ ∈ G∞ . Similarly, replacing the role in this argument of H∞ by H2 and Theorem 2.3.8 by Theorem 2.3.4, we get that Φ◦Ψ∈ G whenever Φ,Ψ∈ G. For the second part of the statement, assume that Φ( s ) = cΦs + φ ( s )and Ψ( s ) = cΨs + ψ ( s ), where cΦ, cΨ∈N∪{ 0 } and φ, ψ ∈ D . Since we already know that Φ◦Ψ∈ G∞, we can write Φ◦Ψ(s) = cΦ◦Ψs+η(s), cΦ◦Ψ∈N∪{0}and η∈ D.(3.3) 61 3. Semigroups of composition operators on Hardy spaces of Dirichlet series But also Φ◦Ψ(s) = cΦΨ(s) + φ(Ψ(s)) = cΦcΨs+cΦψ(s) + φ(Ψ(s)).(3.4) Identifying (3.4) and (3.3), we find that 0=(cΦcΨ−cΦ◦Ψ)s+cΦψ(s) + φ(Ψ(s)) −η(s). Dividing by sin the latter identity, we find that 0 = cΦcΨ−cΦ◦Ψ+1 s(cΦψ(s) + φ(Ψ(s)) −η(s)). If we let Re ( s ) → + ∞ , by Lemma 2.1.9, the term 1 s ( cΦψ ( s ) + φ (Ψ( s )) −η ( s )) converges to 0. This yields the desired conclusion. ■ Throughout this chapter we will be working with semigroups of analytic functions { Φ t} in the classes G and G∞ . For convenience, we shall write from now on ct instead of cΦt . The next result studies the behaviour of the mapping t7→ ct . As we are about to see, the continuous semigroup structure forces this mapping to be necessarily constantly equal to 1. Proposition 3.1.5. Let { Φ t} be a continuous semigroup of analytic functions in the class G∞ . Then, the sequence of symbols {ct}t≥0 is constantly equal to 1. Proof. By Proposition 3.1.4, the characteristic of Φ t◦ Φ u ( s ) = Φ u+t ( s )satisfies cu+t = cuct . We claim that this implies that, for every t≥ 0, ct∈ { 0 , 1 } . Let us assume the existence of t > 0such that ct≥ 2. Let m∈N , m≥ 2. Now, consider Φ t ( s ) = ct + φt ( s )and Φ t/m ( s ) = ct/ms + φt/m ( s ), ct/m ∈N∪{ 0 } , φt, φt/m ∈ D . By the semigroup structure, clearly, Φ t = Φ t m◦···m◦ Φ t m = Φ t mm . Then, applying Proposition 3.1.4 m-times, we find that ct= (ct m)m.(3.5) If ct/m ∈ { 0 , 1 } , then so is ct and we reach a contradiction. Therefore, suppose that ct/m ≥ 2, for all m . If this occurs, since m is arbitrary and ct is a natural, (3.5) can only hold if ct∈ { 0 , 1 } , which contradicts our initial assumption on ct . This gives the claim. In order to exclude the case ct = 0, by the continuity of the semigroup, the functions Φ t must be univalent (Theorem 1.1.7). Nonetheless, by Theorem 3.1.1, Dirichlet series fail to be so. Consequently, ct = 1 for every t≥ 0, as desired. ■ Remark 3.1.6.If the functions of the semigroup belong to the class G , there is an alternative way to conclude the above proof without using the univalence of the functions of the semigroup. Indeed, take s0> 0such that cs0 = 0. Then ct = ct−s0cs0 = 0 for every t≥s0 . Note that this implies that whenever cu = 1, then ct = 1 for every t≤u . Therefore, there is a point t0 such that ct = 1 if t < t0 and ct = 0 if t > t0 . We claim that t0 = ∞ . Assume first that 0< t0<∞. Then c4 3t0=c2 3t0c2 3t0= 1. 62 3.2. Semigroups of composition operators However, this contradicts the fact that ct = 0 for any t > t0 . Now, if t0 were equal to zero, it would mean that ct = 0 for every t > 0. Now, since Φ t→ Φ 0 as t→ 0on compact sets of C+ , we would have that Φ t (1 / 4) → 1 / 4as t→ 0. However, since Φ t∈ G , we know that Re (Φ t (1 / 4)) > 1 / 2, in contradiction with our last statement. Hence, t0=∞and, consequently, ct= 1 for every t≥0. Remark 3.1.7.Notice that in the proof of Proposition 3.1.5 we only use the continuity of the semigroup in the last part of it. In particular, this implies that the fact of having a semigroup { Φ t} in the class G already forces ct∈ { 0 , 1 } . An example of such a semigroup is the one given by Φ 0 ( s ) = s , s∈C+ ,Φ t ( s ) = s0 , with s0∈C1/2 , for all t > 0. Clearly, Φ t∈ G , for all t≥ 0, and { Φ t} is a semigroup. But, in this case, c0= 1 and ct= 0 for all t > 0. Remark 3.1.8.We now consider a non-continuous semigroup { Φ t} in the class G , which, in contrast with the previous example, satisfies that ct = 1 for all t≥ 0. For example, take a non-continuous function f : R→R such that f ( t + u ) = f ( t ) + f ( u )for all t, u ∈R (see [ 35 , Example 8.1.13]) and consider Φ t ( s ) = s + if ( t ) . Clearly, { Φ t} is a non-continuous semigroup in G and ct = 1 for all t≥0. Remark 3.1.9.In Chapter 2, it was shown that G⊊G∞ . However, when dealing with continuous semigroups of analytic functions in the class G , these classes coincide. Indeed, we recall that the class G consists on those elements Φin G∞ such that whenever the characteristic cΦ is zero, Φ( C+ ) ⊂C1/2 . However, by Proposition 3.1.5, every element of a continuous semigroup has characteristic equal to 1. So in a continuous semigroup { Φ t} in G∞ there can be no elements in G∞\G. Proposition 3.1.10. Let { Φ t} be a non-trivial continuous semigroup in the class G∞ . Then, the maps Φ t are parabolic self-maps of C+ and the Denjoy-Wolff point of the semigroup is ∞. Proof. If { Φ t} is a semigroup in the class G∞ , by Proposition 3.1.5, Re Φ t ( s ) ≥ ctRes = Res for all s in C+ . This clearly implies that Φ t has no fixed point in C+ and, in fact, by Theorem 1.2.1, the Denjoy-Wolff point cannot be a finite number, that is, lim t→+∞Φt(s) = ∞, s ∈C+.(3.6) Moreover, for every t > 0, by Lemma 2.1.9 and Proposition 3.1.5, lim Re s→+∞ Φt(s) s= 1. With the standard classification of dynamics, this means that each function Φ t is a parabolic self-map of C+ (see Remark 1.2.4 for the classification in the setting of the right half-plane). ■ 3.2 Semigroups of composition operators The theory of strongly continuous semigroups of bounded operators on Banach spaces has been a fruitful tool in a great number of areas in Analysis. Let us recall this notion. 63 3. Semigroups of composition operators on Hardy spaces of Dirichlet series 3.3 The infinitesimal generator In the study of both semigroups of operators and of holomorphic functions, the infinitesimal generators play a fundamental role. See, i.e., [ 35 , Chapter 10] for the case of holomorphic semigroups. Regarding semigroups of operators we refer the reader either to [ 53 , Chapter II] or [ 80 , Chapter 13]. The aim of this section is to characterise the infinitesimal generators of continuous semigroups in the Gordon-Hedenmalm class. As a byproduct, we will describe the infinitesimal generator of a strongly continuous semigroup of composition operators in Hardy spaces of Dirichlet series. Let us recall that given a Banach space X and {Tt} an operator semigroup where Tt:X→X, the infinitesimal generator of the semigroup is defined as Af = lim t→0+ Ttf−f t,(3.11) where the convergence is considered in the norm topology. We denote by D ( A ) the set of all f∈Xsuch that the limit (3.11) exists. A classical result from general semigroup theory guarantees that if the semigroup of composition operators {Tt} is strongly continuous, then D ( A )is dense in the space X(see, i.e. [53, p.37, Theorem 1.4]). 3.3.1 Infinitesimal generators of continuous semigroups in G In Theorem 3.2.6, it was established an analogue version of Berkson-Porta’s Theorem in the setting of Hardy spaces of Dirichlet series. Nonetheless, at this point, the natural question of finding examples of continuous semigroups { Φ t} in the class Garises. Clearly, the translations Φt(s) = s+at, Rea≥0 are a straightforward example of such semigroups. The present section is devoted to the description of the infinitesimal generators of the continuous semigroup in the Gordon-Hedenmalm class. Such description will show the existence of a rich variety of examples of continuous semigroups in G. As it was seen in Proposition 3.1.10, the Denjoy-Wolff point of a semigroup in the Gordon-Hedenmalm class is ∞ . Thus, Theorem 1.3.4 ensures that its infinitesimal generator is a holomorphic function sending the right half-plane into its closure. Clearly, not every holomorphic function H : C+→C+ is necessarily the infinitesimal generator of a continuous semigroup in the class G : Example 3.3.1. Let H : C+→C+ be H ( z ) = z . A direct computation shows that Φ t ( z ) = zet , z∈C+ , t≥ 0. Since the identity is not in D , the semigroup {Φt}is not in G. Taking this into account, the natural question arising is what extra assumptions should an infinitesimal generator H satisfy so that the elements of the resulting continuous semigroup in C+ belong to the class G . The main result of this section is the following characterisation of the infinitesimal generators of continuous semigroups in the Gordon-Hedenmalm class. 70 3.3. The infinitesimal generator Theorem 3.3.2. Let H : C+→C+ be analytic. Then, the following statements are equivalent: a) H is the infinitesimal generator of a continuous semigroup of elements in the class G. b) H∈ H∞(Cε),for all ε > 0. c) H∈ D. This is, the infinitesimal generator of a continuous semigroup { Φ t} in the class G is a holomorphic function H : C+→C+ which belongs to the Fréchet space introduced by Bonet in [33]. The proof of this result will be given at the end of this subsection (see page 76). In fact, it will be an easy consequence of some more general results. Theorem 3.3.3. Let Λbe a multiplicative semigroup of N and { Φ t} a continuous semigroup in G such that φt∈ DΛ . Let H be its infinitesimal generator. Then, H∈ H∞ Λ(Cε),ε > 0. Proof. By Theorem 1.3.1, we have that H(s) = lim t→0+ Φt(s)−s t= lim t→0+ φt(s) t, where the limit is uniform on compact subsets of C+ . Now, notice that for every t > 0, the functions gt ( s ) = φt ( s ) /t belong to the class G∞ and, therefore, they are bounded in every half-plane Cε . By Theorem 2.4.2, we actually have that the limit is uniform on half-planes Cε , ε > 0. This implies that H is an element belonging to G∞ with characteristic equal to zero and, in addition, we have that H∈ H∞ ( Cε ), ε > 0. The membership of H to DΛ follows from the fact that the uniform convergence in Cε implies the continuity of the coefficients. Hence, being the series gtelements in DΛ, so is the limit series H.■ There exists a less direct way to prove Theorem 3.3.3 without passing through Theorem 2.4.2. We shall give this alternative argument since it provides some useful information concerning the domains of the infinitesimal generators of the strongly continuous semigroups of composition operators in Hp ,1 ≤p < ∞ . We find convenient to detail the argument straightaway. For clarity, we extract the following lemma for the alternative proof of Theorem 3.3.3. The proof of this lemma is similar to the one of [ 79 , Lemma 10.29]. Lemma 3.3.4. Let f∈ H2 and 1 / 2 < α < β . Consider the vertical strip Ω = {s∈C:α < Re(s)< β}and define F: Ω ×Ω→Cby F(z, w) =      f(z)−f(w) z−wif z=w, f′(z)if z=w. (3.12) Then, the function Fis uniformly continuous on Ω×Ω. 71 3. Semigroups of composition operators on Hardy spaces of Dirichlet series Proof. Choose ε > 0. By Lemma 2.2.18, there is M > 0such that |f′′ ( s ) | ≤ M for all s∈ Ω. Take δ = ε/M and z0, z1∈ Ωsuch that |z0−z1|< δ . Consider the segment γ(t) = tz0+ (1 −t)z1, t ∈[0,1]. Then, |f′(z0)−f′(z1)|=Z1 0 f′′(γ(t))(z0−z1)dt≤M|z0−z1|< ε. Now, take ( z0, w0 ) , ( z1, w1 ) ∈ Ω × Ωsuch that |z0−z1|< δ and |w0−w1|< δ . Notice that |F(z0, w0)−F(z1, w1)|=Z1 0 (f′(w0+t(z0−w0)) −f′(w1+t(z1−w1)))dt ≤Z1 0|f′(w0+t(z0−w0)) −f′(w1+t(z1−w1))|dt <ε, since |w0 + t ( z0−w0 ) −w1−t ( z1−w1 ) | ≤ δ , for all t∈ [0 , 1], and we are done. ■ Second proof of Theorem 3.3.3. By ‘(i) implies (ii)’ in Theorem 1.5.18 , it suffices to show that the function H is bounded in some half-plane Cε , for some ε > 0. We will show that His bounded in C1/2+σ, for σ > 0fixed. Hence, fix σ > 0. By Lemma 2.2.18, there is a constant C ( σ ) > 0such that |f(s)| ≤ C(σ)∥f∥H2and |f′(s)| ≤ C(σ)∥f∥H2for all f∈ H2and s∈Cσ+1/2. Given ε > 0and h ( s )=2 −s , take f∈ H2 a function in the domain of the infinitesimal operator of the strongly continuous semigroup Tt = CΦt in H2 such that ∥f−h∥H2< ε. Moreover, we have |f(s)−h(s)| ≤ C(σ)ε(3.13) and |f′(s)−h′(s)| ≤ C(σ)ε, (3.14) for all s∈C1/2+σ . Thus, using (3.13) and (3.14) for σ/ 2and ε small enough, both |f| and |f′| are bounded below by a positive constant in the vertical strip Ω = {s∈C:1 2+σ 2<Re(s)<1 2+ 3σ} ⊂ C1/2+σ/2. Take the function F introduced in Lemma 3.3.4 associated with the function fand consider the function K(z, w) =      2−z−2−w z−w, if z=w, −ln(2)2−z,if z=w. Then, given z, w ∈Ω, |F(z, w)−K(z, w)| ≤ Z1 0|f′(w+t(z−w)) −h′(w+t(z−w))|dt ≤C(σ/2)∥f−h∥H2≤C(σ/2)ε. (3.15) 72 3.3. The infinitesimal generator Let us give a lower bound of |K|, |K(z, w)|= 2−z−2−w z−w= 2−Re(z)|1−2z−w| |z−w| Note that | 1 − 2 u|/|u| tends to ln 2as |u| → 0. Thus, there is δ0> 0such that |K(z, w)| ≥ 2−Re(z)ln 2 2, whenever |z−w| ≤ δ0 . Therefore, there exists C > 0such that |K ( z, w ) |> C for all z, w ∈ Ωwith |z−w| ≤ δ0 . Taking ε≤C 2C(σ/2) , (3.15) necessarily forces |F(z, w)|> C/2for any z, w ∈Ωsuch that |z−w| ≤ δ0. By Theorem 3.2.6,Φ t converges to the identity map uniformly in Ω. Hence, for t small enough and 1 / 2 + σ/ 2 <Res < 1 2 + 2 σ , it holds Φ t ( s ) ∈ Ωand | Φ t ( s ) −s|< δ0 . Therefore, |F (Φ t ( s ) , s ) | ≥ C/ 2and limt→0F (Φ t ( s ) , s ) = F ( s, s ) = f′ ( s )uniformly in Ω 1 = {s∈C : 1 2 + σ < Re ( s ) <1 2 + 2 σ} . Thus limt→01 F(Φt(s),s)=1 f′(s)uniformly in Ω1. By the very definition of infinitesimal generator, there exists g∈ H2 such that lim t→0 f(Φt(s)) −f(s) t=g(s)(3.16) in the norm of H2. Then lim t→0 f(Φt(s)) −f(s) t=g(s), uniformly on the vertical strip Ω1. The introduction of the function F allows us to rewrite the incremental quotient in (1.1) as gt(s) := φt(s) t=Φt(s)−s t=f(Φt(s)) −f(s) t·1 F(Φt(s), s). Putting all together, and using that both g and 1 f′ are bounded on Ω 1 , we have H(s) = lim t→0+ Φt(s)−s t=g(s) f′(s), uniformly on the vertical strip Ω 1 . H is a holomorphic function in C+ and gt converges uniformly on Ω 1 , and then in C1/2+σ , to H . Thus H∈ H∞ ( C1/2+σ ) for every σ > 0. In the case φt∈ DΛ for all t , we obtain that gt∈ H∞ Λ ( C1/2+σ ) , for all t , and thus H∈ H∞ Λ(C1/2+σ).■ Remark 3.3.5.As it was mentioned prior to the proof, a byproduct of this second proof of Theorem 3.3.3 is the following: for every f∈ H2∩D ( A ), where A is the infinitesimal generator operator of the semigroup {Tt} , Tt = CΦt , we have that Hf′∈ H2 . In fact, the proof for the case p = 2 can be easily adapted for the case 1 ≤p < ∞ . To do so, it suffices to consider at the beginning of the proof Remark 2.2.19. The rest of the proof works in a same fashion. This consequence will allow us to compute in Section 3.3.2 the infinitesimal generator of the strongly continuous semigroups in Hp,1≤p < ∞. 73 3. Semigroups of composition operators on Hardy spaces of Dirichlet series The converse of Theorem 3.3.3 requires the next technical theorem. Following the ideas in the Picard-Lindelöf Theorem, for σ > 0and δ > 0small, we consider the space X consisting on the collection of functions f : C1+σ× [0 , δ ] →C1+σ satisfying the following three properties i) fis continuous on C1+σ×[0, δ]; ii) s7→ f(s, t)−s∈ H∞(C1+σ)for each t∈[0, δ]; iii) The map [0, δ]→ H∞(C1+σ)given by t7→ (s7→ f(s, t)−s), is continuous. Notice that X depends on σ and δ but we do not write explicitly such dependence in order to simplify the exposition. We endow X with the distance defined for f, g ∈Xas d(f, g) = ∥f−g∥∞= sup s∈C1+σ, t∈[0,δ] |f(s, t)−g(s, t)|.(3.17) Note that conditions ii) and iii) guarantee that d : X×X→ [0 ,∞ ). In fact, ( X, d )is complete. Indeed, let {fn}n be a Cauchy sequence of elements in X . Since |fn(s, t)−fm(s, t)| ≤ d(fn, fm) for all s and t , we have that the sequence {fn ( z, t ) }n is Cauchy in C . This guarantees the existence of the limit f(s, t) := lim n→∞ fn(s, t). A standard argument shows that this convergence is uniform in C1+σ× [0 , δ ], so that f is continuous. Regarding the second property, the uniform limit of bounded Dirichlet series in C1+σ yields again a bounded Dirichlet series in the same half-plane. Eventually, t7→ f ( z, t ) −z maps the interval [0 , δ ]into the algebra H∞ ( C1+σ )and, being f the uniform limit of continuous f , the map is continuous too. Therefore, the metric space (X, d)is complete. Proposition 3.3.6. Let H : C+→C+ be analytic and such that H∈ H∞(C1/2+σ)for every σ > 0. We define the operator Tin Xgiven by Tf(s, t) = s+Zt 0 H(f(s, τ))dτ, f ∈X, (s, t)∈C1+σ×[0, δ]. Then, there is δ=δ(H, σ)small enough, such that 1) T:X→X, 2) Tis contractive. Proof. Observe that for s∈C1/2+2σ , the uniform convergence of the Dirichlet series defining Hallows us to write −H′(s) = ∞ X n=1 anlog n n−s. 74 3.3. The infinitesimal generator Now, since H∈ H∞ ( C1/2+σ ), by the Cauchy integral formula, we obtain that H′is bounded in C1+σfor every σ > 0. Fix σ > 0. Take M= max{sup s∈C1+σ|H′(s)|,sup s∈C1+σ|H(s)|} and δ < 1/M. Let us see first that Tf maps C1+σ× [0 , δ ]into C1+σ . This is indeed the case because Re(Tf(s, t)) = Re(s) + Zt 0 Re(H(f(s, τ)))dτ > 1 + σ, s ∈C1+σ, t ∈[0, δ], where we have used that Re ( H ) ≥ 0. For the continuity, notice that, given s∈C1+σ,0< t < t0< δ, |Tf(s, t)−T f(s0, t0)| ≤ |s−s0|+Zt 0 H(f(s, τ))dτ −Zt0 0 H(f(s0, τ))dτ ≤ |s−s0|+Zt 0|H(f(s, τ)) −H(f(s0, τ))|dτ +Zt0 t|H(f(s0, τ))|dτ ≤ |s−s0|+M|t−t0|+MZt 0|f(s, τ)−f(s0, τ)|dτ. From these inequalities and the very definition of X , we deduce that Tf is continuous in C1+σ×[0, δ]. Now, we verify that s7→ Tf ( s, t ) −s = Rt 0H ( f ( s, τ )) dτ belongs to H∞ ( C1+σ ). In virtue of Theorem 2.3.3 we deduce that H◦f ( ·, τ ) ∈ D for every τ and since H∈ H∞ ( C1/2+σ ), we have that F ( ·, τ ) = H◦f ( ·, τ ) ∈ H∞ ( C1+σ ). Moreover, using again that H′ is bounded, we deduce that the function τ7→ F ( ·, τ )is continuous. Thus, using that H∞ ( C1+σ )is a Banach space, we have that Rt 0H ( f ( ·, τ )) dτ also belongs to H∞ ( C1+σ )and that the map t7→ Rt 0H(f(·, τ))dτ is continuous. Thus, we have obtained 1). For the contractivity, let f1, f2∈X. Then, given s∈C1+σ,t∈[0, δ] |Tf1(s, t)−Tf2(s, t)| ≤ Zt 0|H(f1(s, τ)) −H(f2(s, τ))|dτ ≤MZt 0|f1(s, τ)−f2(s, τ)|dτ ≤tM∥f1−f2∥∞=tMd(f1, f2)≤δMd(f1, f2). Since δM < 1, we get the contractivity. ■ Theorem 3.3.7. Let H : C+→C+ be analytic and such that H∈ H∞ ( C1/2+σ ) for every σ > 0. Then, H is the infinitesimal generator of a continuous semigroup {Φt}where Φt(s) = s+φt(s) and φt∈ D , for all t , that is, { Φ t} is a continuous semigroup in the GordonHedelmann class G. 75 3. Semigroups of composition operators on Hardy spaces of Dirichlet series Proof. We notice that whenever the boundary of C+ is attained, then H is constant and the result is straightforward. Thus, we assume that H : C+→C+ . By Berkson and Porta’s Theorem (see Theorem 1.3.4), there exists a unique continuous semigroup { Φ t} in C+ such that H is its infinitesimal generator. In particular, the map [0 , + ∞ ) ∋t7→ Φ t ( s )is the unique solution of the Cauchy problem ∂Φt(s) ∂t =H(Φt(s)) and Φ0(s) = s∈C+. On the other hand, and following the notation introduced in Proposition 3.3.6, by the Banach Fixed Point Theorem, there are δ > 0, a continuous function f:C2×[0, δ]→C2satisfying the following three properties i) s7→ f(s, t)−s∈ H∞(C2)for each t∈[0, δ]; ii) the map given by t7→ f(·, t)−·, from [0, δ]to H∞(C2)is continuous; iii) and f is a fixed point of the operator introduced in Proposition 3.3.6. That is f(s, t) = s+Zt 0 H(f(s, τ))dτ, f ∈X, (s, t)∈C2×[0, δ]. Write gt ( s ) = f ( s, t )for s∈C2 and t∈ [0 , δ ]. Clearly, the map [0 , δ ) ∋t7→ gt ( s ) is a solution of the Cauchy problem ∂gt(s) ∂t =H(gt(s)) and g0(s) = s∈C2. Thus, by the uniqueness of the Cauchy problem, Φ t ( s ) = gt ( s )for 0 ≤t<δ and Res > 2. In particular, this implies that s7→ Φ t ( s ) −s is a Dirichlet series for t<δ . That is, for those values of t ,Φ t∈ G . Finally, by Proposition 3.1.4 and the very definition of semigroup, we have that Φ t ( s ) = Φ t mm ( s ), with t/m < δ for m∈Nlarge enough. Then, we deduce that Φt∈ G for all t > 0.■ Remark 3.3.8.If the function H in Theorem 3.3.7 belongs to H∞ Λ ( C1/2+σ ), with Λa multiplicative semigroup of natural numbers, our proof can be easily adapted to get that φt∈ DΛfor all t > 0. Proof of Theorem 3.3.2. For a )implies b ), we use first Theorem 3.3.3 and then Theorem 3.1.2. The fact that b )implies c )is trivial. For c )implies a ), we use Theorem 3.1.2 and we find that σu ( H ) ≤ 0. This allows us to apply Theorem 3.3.7 and a)follows. ■ Notice that Theorem 3.3.2 provides a way to build examples of continuous semigroups { Φ t} in the class G . Indeed, we have to consider a holomorphic funcion H : C+→C+ (we exclude the case where H touches the imaginary axis iR , since it corresponds to the elementary examples) which can be represented as a convergent Dirichlet series in some half-plane Cη , η∈R . Then, the semigroup will be obtained by solving the Initial Value Problem ∂Φt(s) ∂t =H(Φt(s)) and Φ0(s) = s∈C+. 76 3.3. The infinitesimal generator In Example 3.5.10 we obtain a continuous semigroup in the Gordon-Hedenmalm class following this method. One way to provide examples of continuous semigroups is the following. Consider a holomorphic function G : D→C+ , with G ( z ) = P∞ n=0 anzn , z∈D . Fix an integer q≥2and define H(s) = G(q−s) = ∞ X n=0 an(qn)−s, s ∈C+. By Theorem 3.3.7, there exists a continuous semigroup { Φ t} in G such that H is its infinitesimal generator. Moreover, by the previous remark, as Λ = {qn : n≥ 0 } is a multiplicative semigroup, it is easy to deduce that for every t≥ 0there exists gt : D→C+ holomorphic such that Φ t ( s ) = s + gt ( q−s ), s∈C+. 3.3.2 Infinitesimal generators of strongly continuous semigroups of composition operators in Hp Let 1 ≤p < ∞ and take a strongly continuous semigroup of composition operators {Tt} given by Ttf = f◦ Φ t , where { Φ t} is a continuos semigroup in the class G . Denote by A the infinitesimal generator of {Tt} . By Theorem 3.3.2, the infinitesimal generator of { Φ t} is a Dirichlet series H sending C+ in C+ . Take f∈ D(A). Then, by the very definition of Aand the chain rule Af(s) = lim t→0+ f◦Φt(s)−f(s) t=f′(s)∂ ∂t(Φt(s))t=0 =f′(s)H(s) whenever Re s > 1/2. Moreover, by Remark 3.3.5, we have that D(A)⊂ {f∈ Hp:Hf′∈ Hp}. The other inclusion was proved in [ 25 , Theorem 2] in a much more general context using properties of the resolvent of a semigroup of operators. Thus, we have: Proposition 3.3.9. Let 1 ≤p < ∞ and take a strongly continuous semigroup of composition operators Ttf = f◦ Φ t in Hp . Then, there is a Dirichlet series H : C+→C+ such that the infinitesimal generator is given by the operator A(f) = Hf′and D(A) = {f∈ Hp:Hf′∈ Hp}. 3.3.3 Uniformly continuous semigroups in Hpspaces From the very beginning of this chapter, we have been working with strongly continuous semigroups. Another standard and useful notion of semigroups of bounded operators is that of uniformly continuous semigroup. Let us recall this notion. Definition 3.3.10. Consider a Banach space X and a semigroup of bounded operators {Tt} in X . It is said that {Tt} is uniformly continuous if and only if Tt converges, as t goes to 0, to the identity map in the norm of the space of bounded operators in X. More precisely: lim t→0+∥Tt−Id∥= 0, 77 3. Semigroups of composition operators on Hardy spaces of Dirichlet series where Id stands for the identity operator in Xand ∥·∥is the operator norm. Clearly, every uniformly continuous semigroup is strongly continuous. A classical result states that a strongly continuos semigroup with infinitesimal generator A is uniformly continuous if and only if A is bounded in X and if and only if D ( A ) = X . In such a case, Tt = etA for all t > 0. See, i.e. [ 53 , Corollary 1.5, Page 39]. We will show that no non-trivial semigroup of composition operators is uniformly continuous on Hp. Theorem 3.3.11. 1 ≤p < ∞ . Let {Tt} be a uniformly continuous semigroup of composition operators in Hp. Then, Tt= Id for every t≥0. Proof. By Proposition 3.3.9, if the semigroup is strongly continuous with infinitesimal generator A, then D(A) = {f∈ Hp:Hf′∈ Hp}, where H : C+→C+ is a Dirichlet series. Assume that A is bounded. Notice that, for each n≥ 2, the operator Mn : Hp→ Hp given by Mn ( f ) = n−sf is an isometry. Using this, we have that ∥A(n−s)∥Hp=∥H(n−s)′∥Hp= log n∥Hn−s∥Hp= log n∥H∥Hp. Since ∥n−s∥Hp = 1 for all n , we deduce that the operator A is bounded in Hp if, and only if, H≡0. Clearly, this forces Tt=Id for every t≥0, as desired. ■ 3.4 Semigroups of composition operators in H∞ The aim of this section is to prove the following theorem. Theorem 3.4.1. Let {Tt} be a strongly continuous semigroup of composition operators in H∞. Then, Tt= Id for every t≥0. The proof of the theorem is inspired in the one given in [ 78 , Theorem 4.25]. There, the setting is H∞ ( D ), and it is given a new proof of the fact that the only strongly continuous semigroup of composition operators in H∞ ( D )is the trivial one. In order to prove the result, we need a preliminary lemma whose proof is a slight adaptation of “a) implies b)” in Theorem 3.2.6 so we omit it: Lemma 3.4.2. Let { Φ t} be a semigroup in G∞ . If {Tt} = {CΦt} is a strongly continuous semigroup in H∞, then {Φt}is a continuous semigroup in G. Proof of Theorem 3.4.1. Write {Tt} = {CΦt} . By Lemma 3.4.2, { Φ t} is a continuous semigroup in C+ . Therefore, it has an infinitesimal generator H : C+→C+ . H is an holomorphic function in C+ . Assume that H is non-zero. Take L a linear fractional map sending the unit disc onto the right half-plane. Then H◦L is a holomorphic function from D into C+ . Then it has non-tangential limit at almost every point in the boundary of the unit disc and, since it is non-zero, there is a point in the boundary of the unit disc such that the limit is a complex number different from zero (see, i.e., [ 51 , Theorems 3.2 and 2.2]). Then, there are y∈R,ε > 0, and δ > 0such that |H(x+iy)|> δ, (3.18) 78 3.4. Semigroups of composition operators in H∞ for all x∈(0, ε). The standard argument already used in Subsection 3.3.2 shows that the infinitesimal generator of the semigroup {Tt} is given by A ( f ) = Hf′ and, arguing as we did in the proof of Proposition 3.3.9, its domain is D(A) := {f∈ H∞:Hf′∈ H∞}. We claim that for every f∈D(A)there exists lim x→0+f(x+iy). Let us proof this claim. Take f∈D ( A ). Therefore, there exists M > 0such that |H(s)f′(s)|< M, s ∈C+. Now, given s = x + iy such that x∈ (0 , ε ), putting together this bound and (3.18), δ|f′(s)|< M. Then, sup x∈(0,ε)|f′(x+iy)| ≤ M δ.(3.19) This estimate will allow us to apply the Dominated Convergence Theorem to f(x+iy) = f(ε+iy)−Zε x f′(u+iy)du. (3.20) Indeed, (3.19) and the fact that χ(x,ε)→χ(0,ε) as x→ 0 + allows us to apply the Dominated Convergence Theorem to the integral in (3.20) . Taking the limit when x→0+, we find that lim x→0+f(x+iy) = lim x→0+f(ε+iy)−Zε x f′(u+iy)du =f(ε+iy)−Zε 0 f′(u+iy)du. Therefore, the above claim holds. Take now a function f∈D(A)∥·∥H∞. Again we have that lim x→0+f(x+iy) exists. Let us assume on the contrary that such limit does not exist. Then, there would exist two sequences of positive real numbers {un}n∈N and {vn}n∈N both tending to zero and λ1, λ2∈C,λ1=λ2, such that lim n→∞ f(un+iy) = λ1and lim n→∞ f(vn+iy) = λ2. Set κ = 1 3|λ1−λ2|> 0. Since f∈D(A)∥·∥H∞ , there exists g∈D ( A )such that ∥f−g∥H∞< κ. 79 4. Composition operators on the algebra of Dirichlet series that Lefèvre proved in [ 14 ] that they coincide with the weakly compact composition operators on H∞ . We can adapt the arguments used in the A ( C+ )setting, to go beyond this equivalence by establishing an analogue characterisation as the one proved for A ( C+ )by showing that every non-compact composition operator on H∞fixes a copy of ℓ∞(see Theorem 4.3.6). With the description of bounded composition operators settled in Section 4.4 we focus our attention on the study of semigroups of composition operators on the algebra A ( C+ ). Essentially, as in Theorem 3.2.8, we aim to prove an analogue of Berkson and Porta’s result in the setting of the algebra of Dirichlet series. Using that a continuous semigroup { Φ t} in the class G∞ is of the form Φ t ( s ) = s + φt ( s ), φt∈ D , (see Proposition 3.1.5) a one-to-one correspondence between strongly continuous semigroups of composition operators in A ( C+ ) and continuous semigroups of symbols in the class GA is settled in Theorem 4.4.3. In Section 4.4, some properties of the Koenigs functions of the continuous semigroups in the class GA are studied. These are used to provide some interesting examples of continuous semigroups in G∞ . For instance, semigroups in the class G∞which are not in the class GA(see Theorem 4.4.11). The first section of this chapter is devoted to recall some key results from the algebra A ( C+ )established in [ 9 ]. We will also show how this algebra behaves differently to the unit disc algebra A ( D ), the space of analytic functions which are continuous in D . Namely, we provide examples of bounded Dirichlet series, continuous in C+ , but failing to be uniformly continuous in C+ (see Proposition 4.1.7). 4.1 The algebra of Dirichlet series A(C+) We begin by recalling the definition of a new Banach space of Dirichlet series in our exposition. We shall also present some of its most fundamental properties, which will be required throughout the chapter. Definition 4.1.1. We define the algebra of Dirichlet series, denoted by A ( C+ ), as the collection of all Dirichlet series f(s) = ∞ X n=1 ann−s convergent in C+ and such that they define a uniformly continuous function f in C+. The following theorem contains some basic properties of the algebra A ( C+ ), see [9, Proposition 2.2, Theorem 2.3] and [9, Theorem 2.3] for a proof. Theorem 4.1.2. The algebra of Dirichlet series A ( C+ )satisfies the following properties: (1) A ( C+ )is a closed subspace of H∞ . In fact, it is a Banach algebra with the pointwise product of functions. (2) A(C+) = span{n−s:n∈N}∥·∥∞. 86 4.1. The algebra of Dirichlet series A(C+) The proof of Proposition 2.2 in [ 9 ] yields the following more general result on the characterisation of the membership to A ( C+ ). We include its proof in order to recall the ideas used there. Proposition 4.1.3. Let f be a uniformly continuous function in C+ , analytic there and such that f∈ D. Then, f∈ A(C+). Proof. We want to show that the Dirichlet series defining f is in fact bounded in the whole right half-plane and by Bohr’s theorem it also converges there. Since f∈ D , there exists ν∈R such that f is bounded in the closed half-plane Cν by, say, M > 0. We suppose that ν > 0, otherwise there is nothing to prove. By hypothesis, there exists δ > 0such that |f ( s1 ) −f ( s2 ) |< 1for every s1, s2∈C+ such that |s1−s2|< δ . Take k∈N such that 1 k< δ . We consider s∈C+with Re(s)< ν. Then, |f(s)| ≤ k−1 X j=0 fs+j kν−fs+j+ 1 kν+|f(s+ν)| ≤ k+M. Then, the Dirichlet series f is bounded in C+ and, consequently, it converges there. ■ In some sense, Proposition 4.1.3 can be regarded as a sort of Bohr’s theorem for the algebra of Dirichlet series. Indeed, essentially, we can read the statement as given f∈ D having an analytic extension to C+ which is also uniformly continuous, then f∈ A(C+). So we could define the algebra A(C+)as A(C+) = {f:fholomorphic and uniformly continuous in C+}∩D. Example 4.1.4. Consider f(s) = 1 4+2−s+ 3−s+ 5−s, s ∈C+. The function f is clearly in D . Now, 1 /f is uniformly continuous in C+ and it is bounded below there. Therefore, f is uniformly continuous in C+ too. Thus, by Proposition 4.1.3,f∈ A(C+). Remark 4.1.5.In general, in a similar fashion as in H∞ , we can easily provide examples of Dirichlet series in A ( C+ )by considering a function g uniformly continuous in the unit disc Dand taking F(s) = g(k−s),k∈N,k≥2. Example 4.1.6. Let f ( s ) = 1 1+s , s∈C+ . The function f is clearly uniformly continuous in C+ , but it fails to be in D . Indeed, by an analogous argument as the one carried out in Example 2.2.7, in case f were to be representable as a convergent Dirichlet series Pn≥1ann−s, then we would have a1= lim Re s→∞ 1 1 + s= 0 and a2= lim Re s→∞ 2s 1 + s=∞, which clearly constitutes a contradiction. 87 4. Composition operators on the algebra of Dirichlet series The algebra A ( C+ )is another example of how differently Dirichlet series behave with respect to Taylor series in the unit disc D . Indeed, in the unit disc setting, the space of analytic functions which are uniformly continuous in D coincides with those analytic functions having a continuous extension to D . This is no longer true in the case of Dirichlet series. Denote by CE ( C+ ) the class of functions in H∞ having a continuous extension to C+ . Trivially, CE ( C+ )is also a closed Banach subalgebra of H∞ containing A ( C+ ). In [ 70 ], Lefèvre studied composition operators from CE ( C+ )into H∞ . We see now that A(C+)⊊CE(C+). Proposition 4.1.7. There exists a Dirichlet series belonging CE ( C+ )failing to be uniformly continuous in C+. Proof. Let T : D→C+ be the conformal mapping T ( z ) = 1+z 1−z . Define the map φ(s) = 1 22−s−1 23−s,s∈C+. We claim that 1 22−it −1 23−it = 1,for all t∈R. Indeed, if there were a t0∈Rsuch that 2−it0−3−it0= 2, since | 2 −it0| =1= | 3 −it0| , we necessarily would have 2 −it0 = 1 and 3 −it0 = − 1, so t0 = 2 πk/ log 2and t0 = π (2 l +1) /log 3, for k, l ∈Z . In particular, this would imply that log 3 /log 2is a rational number, which constitutes a contradiction. Therefore, there is no real number t such that φ ( it ) = 1. Now, by Kronecker’s Theorem (Theorem 2.2.2), there exists a sequence {tn} of real numbers such that φ ( itn ) → 1as n→ ∞ . Let {hn} be a sequence of positive real numbers such that hn→ 0. Then, we set sn = hn + itn . We claim the existence of a sequence {rn} in C+ such that |rn−sn| → 0as n→ ∞ and, setting zn = φ ( sn ), wn=φ(rn), i)zn→1, ii) T(zn) T(wn)=eπ. Regarding i ), we know that φ ( itn ) → 1. By the choice of the hn ’s, we still have that zn=φ(hn+itn)→1, since φis uniformly continuous in C+. To prove ii ), first notice that since zn→ 1, we have that |T ( zn ) |→∞ , and we can suppose that 2 eπ<|T ( zn ) | . Then, take fsn = sn + εn , where {εn} is a sequence of positive real numbers tending to zero such that for each n 1 22−εn+ 3−εn≤1−2eπ |T(zn)|· Take un=φ(fsn). Clearly, |un|=|φ(fsn)| ≤ (2−εn+ 3−εn)/2. So, |un| ≤ 1−2eπ |T(zn)|· This is, eπ |T(zn)|≤1−|un| 2≤|1−un| |1 + un|=1 |T(un)|· 88 4.1. The algebra of Dirichlet series A(C+) Reordering, we eventually get that  T(zn) T(un)≥eπ. If we take the segments γn ( α ) = αfsn + (1 −α ) sn ,0 ≤α≤ 1, we can find αn∈ (0 , 1) such that |T ( zn ) |/|T ( wn ) | = eπ , where wn = φ ( γn ( αn )). Hence, claim ii)follows with rn=γn(αn). Let A = {z∈C : e−π/2<|z|< eπ/2} . Then, the holomorphic function f ( z ) = exp ( ilog T ( z )) maps D into A . Here log stands for the complex logarithm where the argument is taken in ( −π/ 2 , π/ 2). The Dirichlet series F = f◦φ is clearly convergent in C+ . Since φ can be continuously extended to C+ , 1 ∈ φ ( C+ ), and f is continuous in D\{ 1 } , we conclude that F can be continuously extended to C+ . Regarding the failure of the uniform continuity on C+ , consider the sequences {sn} and {rn} from above. By construction, |sn−rn| → 0as n→ ∞. Now, thanks to part ii)of the claim, we have that |Re(log(T(zn))) −Re(log(T(wn)))|=π. Hence, |Im(ilog(T(zn))) −Im(ilog(T(wn)))|=π. Recalling the definition of f, we find that |Arg(f(zn)) −Arg(f(wn))|=π. Hence, f ( zn )and f ( wn )lie in antipodal segments joining the inner circle and the outer circle of the boundary of A . Therefore, between these two points we can fit, at least, a ball of radius e−π/2 . Therefore, |f ( zn ) −f ( wn ) | ≥ 2 e−π/2 , and the conclusion follows. ■ Remark 4.1.8.The algebra CE ( C+ )is not separable. We can see this using the ideas from the proof of the latter proposition. Let M=2−it :t=(2k+ 1)π log 3 −log 2, k ∈Z. For τ∈∂D , define fτ ( z ) = exp ( ilog T ( τz )), z∈D . Then, we consider Fτ=fτ◦φ, where φ(s) = 1 22−s−1 23−s. We claim the following: i) If τ∈∂D\M, then Fτ∈ CE(C+). ii) For τ, τ′∈∂D\M,τ=τ′: ∥Fτ−Fτ′∥∞≥e−π/2. We begin by showing i). The function fτ fails to be continuous only whenever τz = 1. Hence, so does Fτ in it for those t∈R such that φ ( it ) = τ . If this occurs, we have that, 1 22−it −1 23−it =τ. This is, the middle-point τ of the points 2 −it and − 3 −it in ∂D lies also in ∂D . This implies that 2 −it = − 3 −it = τ . Or, equivalently, 2 it 3 −it = − 1. This forces t=(2k+1)π log 3−log 2 ,k∈Z, and the claim follows. 89 4. Composition operators on the algebra of Dirichlet series Now, we prove ii). Fix τ, τ′∈∂D\M , τ = τ′ . By Kronecker’s Theorem (see Theorem 2.2.2), there exists a sequence {tn} of real numbers such that φ ( itn ) →τ as n→ ∞ . Let {hn} be a sequence of positive real numbers such that hn→ 0. We set sn = hn + itn . Mimicking the construction from the proof of Proposition 4.1.7, we can find a sequence {rn} ∈ C+,φ(rn)→τ, such that |Fτ(sn)−Fτ(rn)| ≥ 2e−π/2.(4.1) Since τ = τ′ , fτ′ can be continuously extended to D\{τ′} and φ ( sn ) , φ ( rn ) →τ , as n→ ∞, we have that |Fτ′(sn)−Fτ′(rn)| → 0.(4.2) Using both (4.1) and (4.2), 2e−π/2≤ |Fτ(sn)−Fτ(rn)| ≤ 2∥Fτ′−Fτ∥∞+|Fτ′(sn)−Fτ′(rn)| for all n∈N. Letting n→ ∞, the claim ii) follows. We have shown that there are uncountable many disjoint open sets in CE(C+), giving the non-separability of the algebra CE(C+). 4.2 Boundedness of the composition operator The condition Φbelonging to G∞ and being uniformly continuous in C+ clearly provides a sufficient condition for the boundedness of the operator CΦ on A ( C+ ). However, it is no longer necessary (see Example 4.2.5). The boundedness of CΦ on A ( C+ )is characterised by the membership of Φto the class GA , which we define right straight-away, where uniform continuity is only required in the sets AM:= {s∈C+: 0 <Re(Φ(s)) < M},for all M > 0. Definition 4.2.1. Let Φ : C+→C+ . We say that Φbelongs to the class GA if Φ∈ G∞and, for every M > 0,Φis uniformly continuous in AM:= {s∈C+: 0 <Re(Φ(s)) < M}. We now prove the main result of the section. Theorem 4.2.2. Let Φ : C+→C+ be analytic. Then, the following statements are equivalent: a) Φdefines a bounded composition operator CΦon A(C+). b) Φ∈ GA. c) n−Φ∈ A(C+)for all n∈N. d) Φ∈ G∞and n−Φ∈ A(C+)for all n∈N. e) Φ∈ G∞and there exists some n≥2such that n−Φ∈ A(C+). Remark 4.2.3.Notice that in the statement of the theorem we are excluding the case in which the image of the symbol Φtouches the boundary of C+ , this is, Φ( C+ ) ∩∂C+ = ∅ . If this happens, then the symbol Φis constant and the composition operator CΦ is trivially bounded on A ( C+ ). Observe also that this is not the case in H∞ since not every H∞ function is defined in the closure of the right half-plane. 90 4.2. Boundedness of the composition operator Proof of Theorem 4.2.2. Trivially, d )implies e ). Now, since n−s∈ A ( C+ )for all n∈N, we clearly have that a)implies c). To show c )implies d ), it is enough to prove that Φ ∈ G∞ . To do so, we use Theorem 2.3.10 and the conclusion immediately follows. Let us show that d )implies a ). By Theorem 2.3.8, CΦ maps H∞ into itself because Φ ∈ G∞ . Moreover, we have by hypothesis that CΦ sends the Dirichlet monomials n−s into A ( C+ ). Since the algebra is the closure of the linear span of the monomials in the H∞ norm (Theorem 4.1.2 (2)) and the operator CΦ is linear, we conclude that CΦ actually maps boundedly the algebra of Dirichlet series A(C+)into itself. For b )implies c ), consider n∈N , n≥ 2, since for n = 1 there is nothing to prove. We want to show that g ( s ) = n−Φ(s) is uniformly continuous in C+ . To do so, take ε > 0and M > 0so that n−(M−1) < ε/ 4. By the uniform continuity of n−s in C+ , there exists δ1> 0such that |n−z−n−w|< ε/ 2 whenever |z−w|< δ1 . By the definition of GA , we have that the symbol Φ is uniformly continuous in AM . Then, there exists a δ > 0so that whenever s1, s2 belong to AM and |s1−s2|< δ , we have | Φ( s1 ) − Φ( s2 ) |< δ1 and thus |n−Φ(s1)−n−Φ(s2)|< ε/2. We may assume that δ < δ1. Now, suppose that neither s1 nor s2 belong to AM−1 . Then, Re (Φ( sj )) > M−1,j= 1,2. Therefore, by the choice of M > 0 |n−Φ(s1)−n−Φ(s2)| ≤ n−Re(Φ(s1)) +n−Re(Φ(s2)) ≤2n−(M−1) < ε/2. Eventually, suppose that s1∈AM−1 and s2∈ AM and |s1−s2|< δ . By the mean value theorem, if γ ( t ) = ts1 + (1 −t ) s2 ,0 ≤t≤ 1, there exists t0 so that M−1<Re(Φ(γ(t0)) < M. Set s3 = γ ( t0 ). Clearly, s1, s3∈AM and |s1−s3|< δ . Therefore, |n−Φ(s1)−n−Φ(s3)|< ε/ 2 . In addition, s2, s3/∈AM−1 and |n−Φ(s2)−n−Φ(s3)|< ε/2. Hence, |n−Φ(s1)−n−Φ(s2)| ≤ |n−Φ(s1)−n−Φ(s3)|+|n−Φ(s3)−n−Φ(s2)|<ε 2+ε 2=ε. We conclude the proof showing e )implies b ). By hypothesis, there exists an n≥ 2such that the Dirichlet series n−Φ belongs to A ( C+ ). Assume that Φ ∈ GA . Since Φ ∈ G∞ , there must exist M0> 1, ε > 0, sequences {rk} , {tk} in AM0such that |rk−tk|<1 kfor all k∈Nand |Φ(rk)−Φ(tk)|> ε, for all k∈N. Consider the rectangle R = {z∈C+ : Re ( z ) ≤ 2 M0,|Im ( z ) | ≤ π 2 log n} . The function f(s) = n−sis bilipschitz there, so, for some constant K=K(M0) |n−s−n−t| ≥ K|s−t|, s, t ∈ R. Let ak = 1 2 ( Im (Φ( rk )) + Im (Φ( tk ))). Hence, if for infinitely many k∈N , the points Φ( rk ) −iak and Φ( tk ) −iak belong to the rectangle R , we would have that |n−Φ(rk)−n−Φ(tk)|=|n−(Φ(rk)−iak)−n−(Φ(tk)−iak)| ≥ Kε. Therefore, n−Φ(s) would not be uniformly continuous in C+ . Then, we can assume that Im (Φ( tk ) − Φ( rk )) > π/ log n for every k . For each k∈N , we 91 4. Composition operators on the algebra of Dirichlet series consider the segments γk ( α ) = αrk + (1 −α ) tk ,0 ≤α≤ 1, joining rk and tk . We now take the image of γk under Φand one of the following two cases must hold: Case I. First, assume, for infinitely many k , the existence of an αk∈ [0 , 1] such that γk ( αk ) ∈ A2M0 . Of course, we may assume that this happens for all k . Set uk = γk ( αk )for which, clearly, |rk−uk|< 1 /k for all k∈N . Then, |n−Φ(uk)|< n−2M0and |n−Φ(rk)|> n−M0(since {rk}is in AM0). Therefore, |n−Φ(uk)−n−Φ(rk)|> n−M0−n−2M0>0. This clearly implies that n−Φ(s) is not uniformly continuous in C+ , a contradiction. Case II. Assume now that the whole image of γk under Φlies in A2M0 for infinitely many k . Again we may assume that this happens for all k . If this were the case, for each kwe could find an αk∈[0,1] so that Im(Φ(γk(αk)) −Φ(rk)) = π/ log n. Set again uk=γ(αk). Hence, Arg(n−Φ(uk))−Arg(n−Φ(rk)) = π, for all k. Since the image of the vertical strip 0 <Re ( s ) < 2 M0 under the function f(s) = n−sis the annulus {z∈D:|z|> n−2M0},we deduce that the modulus of the difference of two points whose argument differs exactly on π units will be strictly greater than 2 n−2M0 . Hence, by the choice we made of both uk and rk, we have that |n−Φ(uk)−n−Φ(rk)|> n−2M0,for all k. Again we have that this implies that n−Φ(s) is not uniformly continuous in C+ , a contradiction. ■ In the following examples we show that the class GA is different from the class G∞ and that the subclass of G∞ of uniformly continuous symbols is strictly contained in GA. Example 4.2.4. The classes GA and G∞ do not coincide. In order to prove this statement, we need to build a symbol Φin G∞ such that, for some M0> 0, Φis not uniformly continuous in AM0 . As it was seen in Definition 2.3.1, the symbols in the class G∞ consist on the analytic functions from C+ to C+ which may be written as cΦs + φ ( s ), where cΦ∈N∪{ 0 } and φ∈ D . We are going to construct a symbol Φwith cΦ = 0. An easy way to obtain a Dirichlet series is through the composition of a holomorphic function on D with a function g ( s ) = n−s , n∈N , n≥ 2, defined on C+ . We take n = 2. Therefore, we consider f∈H∞ ( D ) \A ( D ), where A ( D )stands for the disc algebra, this is, the set of holomorphic functions in D which can be continuously extended to D . Note that f cannot be uniformly continuous in D since, if it were the case, then we could extend it continuously to Dand it would belong to A(D). Now, the function Φ( s ) = f (2 −s ) + ∥f∥∞ is analytic on C+ and it defines a Dirichlet series there. Being clear that f maps C+ into C+ it remains to check that the resulting function is not uniformly continuous in AM0 , for some M0> 0. 92 4.3. Compact composition operators Clearly, Φis not uniformly continuous on C+ . Notice that Φ( C+ )is bounded. So, for M > 0big enough, AM = C+ . Hence, for those M ’s, Φcannot be uniformly continuous in AM , as desired. In brief, the symbol Φinduces a bounded composition operator in H∞but not in A(C+). We now give an example of how the uniform continuity of the symbol in C+ is not a necessary condition in Theorem 4.2.2. Example 4.2.5. There are non-uniformly continuous symbols in C+ which belong to GA .We are going to build Φ ∈ GA which is not uniformly continuous in C+ . Let us consider Φ( s ) = T (2 −s ), where T is a conformal map sending the unit disc onto the intersection of C+ with a horizontal strip. In addition, assume that T(1) = ∞. Clearly, Φ : C+→C+is analytic and it also defines a Dirichlet series there. Therefore, Φ ∈ G∞ . Now, for every M > 0, T−1 ( AM )is contained in the compact set D\E , where E is an open neighbourhood of 1 depending on M . Hence, T is uniformly continuous there. Since f ( s ) = 2 −s is uniformly continuous in the whole C+ , we conclude that Φis in the class GA . Nonetheless, notice that Φcannot be uniformly continuous in C+ since it maps bounded neighbourhoods of the points sn = 2 πin/ log 2, n∈Z , to unbounded regions. In Section 2.4, the closeness of the classes G ∪{ 1 / 2 + iR} and G∞∪{iR} under the local uniform convergence in C+ (or under uniform convergence in half-planes Cε , ε > 0) was proven. Nonetheless, as we are about to see, the class GA is no longer ‘almost’ closed under the local uniform convergence in C+ . The reason being that GA = G∞ (see Theorem 4.2.6 right below) and that there are non-constant symbols in G∞ which do not belong to GA (Example 4.2.4). Theorem 4.2.6. GA=G∞. Proof. As GA⊂ G∞ , we only have to see that G∞⊂ GA . Take Φ ∈ G∞ . Consider the sequence fn ( s ) = Φ( s + 1 /n ), n∈N . Notice that fn∈ GA for every n . Indeed, Φis uniformly continuous in every half-plane Cε , ε > 0(Φ ′ is bounded in every such half-plane), so any horizontal translation Φ τ , τ > 0, is uniformly continuous in C+ . Since fn→ Φuniformly in Cε , we have that Φ∈ GA.■ 4.3 Compact composition operators This section is devoted to the study of the compactness and weak compactness of composition operators in the algebra of Dirichlet series A ( C+ ). Both in H∞ ( D )and in the disc algebra A ( D ), the class of compact composition operators and the class of weakly compact composition operators coincide and can be characterised in terms of the range of the symbol. Namely, if ϕ is such a symbol, the distance of the set ϕ ( D )to ∂D must be positive. In addition, in both spaces the non-compactness is due to the fact that the operator acts as an isomorphism in a non-reflexive subspace, isomorphic to c0 in the disc algebra and to ℓ∞ in the case of H∞ ( D )(see the survey [ 43 ] to see all these results). In this section we will show that the same results hold in our setting. In the case of H∞ , something is known. Namely, Bayart [ 15 ] proved that a bounded composition operator CΦ is compact on H∞ if and only if inf{Re Φ( s ) : s∈C+}> 0and 93 4. Composition operators on the algebra of Dirichlet series Lefèvre [ 70 ] showed that compactness coincides with weak compactness for composition operators on H∞ . This topic was treated in a different context in [10, Proposition 3]. Before starting proving the first auxiliary lemmas, let us recall the following key notion. Definition 4.3.1. Let Ωbe a region in C and {zj} a sequence in Ω. We say that the sequence {zj} is an interpolating sequence in Ω(for H∞ (Ω)), if for every bounded sequence {aj} , there exists a function f∈H∞ (Ω) such that f ( zj ) = aj . This means that the operator T : H∞ (Ω) →ℓ∞ , defined by Tf = f ( zj ), is onto. Given an interpolating sequence {zn} in the unit disc, by Beurling’s Theorem [ 58 , Theorem 2.1, page 285], there exist a constant M > 0and a sequence of functions {gj}in H∞(D)such that gj(zj)=1, gj(zk)=0,if j=k, (4.3) and ∞ X j=1 |gj(z)| ≤ M, for all z∈D.(4.4) In general, it is not possible to get the above functions gj belonging to the disc algebra. Next lemma shows that if the sequence {zn} converges to a point of the boundary, passing to a subsequence {znj} , then the functions can be chosen continuous up to the boundary of the unit disc: Lemma 4.3.2. Let {zn} be a sequence in the unit disc converging to a point τ∈∂D . Then, there exist a subsequence {znj} , a sequence of functions {gj} in the disc algebra A(D), and a constant Msuch that gj(znj)=1, gj(znk)=0,if j=k, (4.5) and ∞ X j=1 |gj(z)| ≤ M, for all z∈D.(4.6) Proof. To simplify the exposition, we assume that τ = 1. Take Ω = D ( − 1 , 2), the disc centred at − 1and radius 2. Notice that the points zn belong to Ω and the sequence {zn} converges to a point of the boundary of Ω. By [ 58 , Theorem 1.1, page 278], there exists a subsequence {znj} which is interpolating for H∞ (Ω). Thus, by Beurling’s Theorem [ 58 , Theorem 2.1, page 285], we can find a constant M1>0and a sequence of functions {hj}in H∞(Ω) such that hj(znj)=1, hj(znk)=0,if j=k, (4.7) and ∞ X j=1 |hj(z)| ≤ M1,for all z∈Ω.(4.8) Notice that, for each j , hj is bounded, analytic in the unit disc and continuous in D\{ 1 } . Now, for each j , take Tj an automorphism of the unit disc such that Tj ( znj )=0and Tj (1) = 1. Finally, consider gj ( z ) = (1 −Tj ( z )) hj ( z ), z∈D . 94 4.3. Compact composition operators The boundedness of hj implies that limz→1gj ( z )=0so that gj∈A ( D )for all j . Using (4.7) and (4.8) , a straightforward computation shows that these functions satisfies (4.5) and (4.6) with M= 2M1.■ We recall the following notion from the theory of Banach spaces. Definition 4.3.3. Given three Banach spaces X , Y , and E , an operator T : X→Y is said to fix a copy of E if there is a subspace Z⊆X such that Zis isomorphic to Eand TZ:Z→T(Z)is an isomorphism. In our case, E will be either the space of bounded sequences ℓ∞ or the space of vanishing sequences c0. Theorem 4.3.4. Let Φ∈ G∞such that inf{Re Φ(s) : s∈C+}= 0. Then 1) CΦ:H∞→ H∞fixes a copy of ℓ∞; 2) CΦ:A(C+)→ H∞fixes a copy of c0. If, in addition, Φ∈ GA, then CΦ:A(C+)→ A(C+)fixes a copy of c0. Proof. We begin by proving 2). By hypothesis, there is a sequence {wj} in C+ such that Re wj goes to zero as j goes to + ∞ , where wj = Φ( sj )for some sj∈C+ . Passing to a subsequence if necessary, we may assume that { 2 −wj} converges to a point τ∈∂D . By Lemma 4.3.2, there exist a subsequence {wnj} , a sequence of functions {gj} in the disc algebra A ( D ), and a constant M such that gj(2−wnj)=1, gj(2−wnk)=0,if j=k, (4.9) and ∞ X j=1 |gj(z)| ≤ M, for all z∈D.(4.10) Write fj ( s ) = gj (2 −s )for all s∈C+ and for all j . The functions fj belong to A ( C+ )for all {αj} ∈ c0 . Take Z the closed subspace in A ( C+ )generated by {fj : j≥ 1 } and take T : c0→Z given by T ( {αj} ) = P∞ j=1 αjfj , for all {αj} ∈ c0 . By (4.10) , T is well-defined. Moreover, T is an isomorphism. Indeed, by (4.10), for all {αj}in c0, ||T({αj})|| ≤ sup w∈C+|X j αjfj(w)| ≤ ||{αj}||∞M and, by (4.9), given j∈N |αj|=|X k αkgk(2−wnj)|=|X k αkfk(wnj)|=|T({αk})(wnj)|≤∥T({αk})∥. (4.11) That is, ∥{αk}∥∞≤ ∥T({αk})∥ ≤ M∥{αk}∥∞ for all {αk} ∈ c0. Moreover, given f=T({αk})∈Z, by (4.11), ∥CΦ(f)∥ ≥ sup j|f(Φ(snj))|= sup j|f(wnj)| ≥ sup j|αj|=∥{αj}∥∞≥ ∥f∥/M. That is, CΦ|Zis an isomorphism. This concludes the proof of 2). 95 4. Composition operators on the algebra of Dirichlet series the unit disc D into the domain Ωsuch that f (0) = 0. By Proposition 4.4.10 a ), there exists a continuous semigroup { Φ t} in G∞ whose Koenigs function is h(s) = −1 log klog(f(k−s)), s ∈C+. Now, by Carathéodory’s Theorem (see [ 35 , Theorem 4.3.1]), since ∂ Ωis not locally connected, f fails to have a continuous extension to ∂D . Then, Proposition 4.4.10 b)gives the conclusion. ■ We can also provide examples of a well known fact for semigroups of analytic functions in the unit disc: there exist continuous semigroups { Φ t} in D such that for some t0> 0the composition operators CΦt are compact in H∞ ( D )for every t>t0but fail to be compact in H∞(D)for t∈[0, t0]. Proposition 4.4.12. There exists a continuous semigroup { Φ t} in GA such that for some t0> 0the composition operators CΦt are compact in A ( C+ )for t>t0 but CΦtfails to be compact in A(C+)for every t∈[0, t0]. Proof. Let D1 = D\ [1 / 2 , 1). Then, D1 is a starlike domain with respect to the origin. Let f : D→D1 be the Riemann mapping fixing the origin and such that f′ (0) > 0. In particular, this implies that f (( − 1 , 1)) = ( − 1 , 1 / 2). Fix c > 0. Now, thanks to Proposition 4.4.10 a ), there exists a continuous semigroup { Φ t} in G∞whose Koenigs function is h(s) = −1 clog 2 log(f(2−s)), s ∈C+. Observe that f can be continuously extended to D (by Carathéodory’s Theorem or by direct verification). This implies, by Theorem 4.4.6, that for every t > 0, Φ t ( s ) = s + φt ( s )is uniformly continuous on horizontal strips of C+ (and so is φt ). By the 2πi log 2 -periodicity of φt , we conclude that for all t > 0,Φ t is uniformly continuous in C+ . Hence, Φ t∈ GA , for every t > 0. Now, h maps C+ into D2 , where D2=C+\Sk∈Z{t+ 2kπi :t∈[0,1/c]}. Suppose that t > 1/c. Then, Φt(C+) = h−1(h(C+) + t)⊂h−1(Ct). We claim that there exists η > 0such that h−1 ( Ct ) ⊂Cη . If this were not the case, there would exists a sequence {sn} in C+ such that Re ( sn ) → 0and Re(h(sn)) > t, for all n. For each n, set zn= 2−sn. Then, |zn| → 1. Now, |f(zn)|= exp(−clog(2)Re(h(sn))) <exp(−ct log 2) = 2−tc <1/2. Nonetheless, since |zn| → 1, necessarily f ( zn ) →∂D1 , which constitutes a contradiction with the fact that |f ( zn ) |< 2 −tc < 1 / 2. Therefore, by Theorem 4.3.5, the operator CΦtis compact for every t > 1/c. Suppose now that t≤ 1 /c . Let S = {z∈C+ : 0 <Im ( z ) < 2 π/ log (2) } . The map ϕ ( s ) = f (2 −s )is a bijection between the set S and D\ [0 , 1). Then, there exists a sequence {sn} in S such that f (2 −sn ) → 1. Therefore, considering the principal branch of the logarithm of f (2 −sn ), we have that h ( sn ) → 0. Hence, h ( sn ) + t→t∈∂D1 as n→ ∞ . Then, h−1 ( h ( sn ) + t ) →∂C+ as n→ ∞ . We conclude that Re (Φ t ( sn )) → 0. By Theorem 4.3.5, the semigroup {Φt}satisfies the desired properties with t0= 1/c.■ 102 4.4. Semigroups of composition operators 4.4.2 Some final examples We begin by giving a condition on the infinitesimal generator H so that the corresponding semigroup {Φt}lies in GA. Lemma 4.4.13. Let Hbe a holomorphic function in C+such that H∈ D and H′∈ H∞. Then, H∈ A(C+). Proof. Since H′ is bounded, H is, in fact, Lipschitz-continuous in C+ and, consequently, uniformly continuous in C+. Hence, H∈ A(C+).■ Proposition 4.4.14. Let H : C+→C+ be in H∞ and such that H′∈ H∞ . Let { Φ t} be the continuous semigroup associated with H . Then, the semigroup {Φt}is in GA. Proof. The statement follows immediately from Gronwall’s Lemma [ 35 , Lemma 10.5.5]. Indeed, set M := ∥H′∥∞ , consider s1, s2∈C+ and let G ( t ) = |Φt(s1)−Φt(s2)|. Then, for t > 0fixed, G(t) = |Φt(s1)−Φt(s2)|≤|s1−s2|+Zt 0|H(Φτ(s1)) −H(Φτ(s2))|dτ ≤G(0) + MZt 0|Φτ(s1)−Φτ(s2)|dτ =G(0) + MZt 0|G(τ)|dτ. Then, Gronwall’s Lemma yields G(t)≤G(0)exp(Mt). Therefore, for each t > 0, the function Φ t is Lipschitz so, in particular, it is uniformly continuous. By Theorem 4.4.3 the conclusion follows. ■ Let us now provide some examples showing how the somehow expected conditions to be satisfied by the infinitesimal generator of a continuous semigroup in the class GA in order to obtain a description similar to the one given in Theorem 3.3.2 do not hold. Throughout the next examples, { Φ t} will denote a continuous semigroup in the class G∞ , H will stand for its infinitesimal generator and hfor its Koenigs function. Example 4.4.15. Φt∈ GAfor all t > 0does not imply that h∈ A(C+). Let h′ ( s ) = 1 1−2−s . By Corollary 3.5.7, its primitive, h is the Koenigs function of a continuous semigroup { Φ t} in G∞ . Now, since H = 1 /h′ is a Dirichlet polynomial, by Proposition 4.4.14, we conclude that {Φt}is actually in GA . However, when s = x , x∈ (0 , ε ), ε > 0, the function h′ behaves like 1 /x and its primitive is clearly unbounded close to zero. Example 4.4.16. Φt∈ GAfor all t > 0does not imply that H∈ H∞. Now, we take h′ ( s ) = 1 + 2 −s . Its primitive is clearly uniformly continuous in C+ . On the other hand, since H = 1 /h′ = g (2 −s ), where g : D→C+ is holomorphic, then the semigroup is of the form Φ t ( s ) = s + ft (2 −s ), where ft : D→C+ for every t > 0. Then, Corollary 4.4.9 guarantees us that Φ t∈ GA . However, H(s)=1/(1 + 2−s)is not in H∞. 103 4. Composition operators on the algebra of Dirichlet series Example 4.4.17. H∈ H∞does not imply that Φt∈ GAfor all t > 0. For n∈N , consider the segments γn = {z∈D : 1 / 2 ≤ |z|< 1 ,arg ( z ) = π 2n} and let γ0 = [1 / 2 , 1). We set Γ = Sn≥0γn . Then, the domain D1 = D\ Γis starlike. Consider f:D→D1 a Riemann mapping fixing the origin. Proposition 4.4.10 guarantees the existence of a continuous semigroup { Φ t} in G∞ whose Koenigs function is h(s) = −1 log 2 log(f(2−s)), s ∈C+. We consider the function h1 ( s ) = h ( s ) + ds , where Re ( d ) > 0. Since Re ( h′ ) > 0 and h′ 1 is still a Dirichlet series mapping C+ into itself, we conclude that h1 is the Koenigs function of a continuous semigroup { Ψ t} in G∞ . Clearly, |h′ 1|>Re ( h′ ) >Re ( d ) > 0, so H1 = 1 /h1∈ H∞ . However, since ∂ Ωis not locally connected, f fails to have a continuous extension to ∂D and, consequently, so does h1 . Theorem 4.4.6 prevents the semigroup { Ψ t} from having a continuous extension to C+and, therefore, to be in the class GA. 104 CHAPTER 5 Evaluation functionals on the spaces Ap α In Chapter 2, we presented the Bergman spaces Ap µ of Dirichlet series, originally introduced by Bailleul and Lefèvre in [ 14 ]. We recall that in this context, µ is a probability measure on (0 , + ∞ )such that 0 ∈supp ( µ ). As it was mentioned there, depending on the choice of the measure µ , the resulting spaces Ap µ might be of special interest. This is the case for the family of measures µα with densities: dµα(σ) = 2α+1 Γ(α+ 1)σαe−2σdσ, α > −1. These measures were originally considered by McCarthy in [ 71 ] for the case p = 2, and ever since they have been object of deeper study (see, for instance, [12] or [55]). In such case, we write Ap αinstead of Ap µα. One of the first properties studied when dealing with Banach spaces of analytic functions on a domain Ω ⊂C is the boundedness of the evaluation functionals δs , s∈ Ω. As it was seen in Chapter 2, the study of these functionals is crucial in the theory of Banach spaces of Dirichlet series. There, it was shown how the boundedness of the evaluations δs is used in the construction of the spaces Hp and Ap µ , respectively. The proof of this boundedness also allows to provide maximum convergence half-planes of the functions belonging to these spaces. Not only proving the boundedness, but computing the norm of δs or, at least, knowing its asymptotic behaviour near the boundary on spaces of analytic functions has many potential applications. For instance, to mention one, it is one of the key ingredients in order to provide necessary conditions for the boundedness of the Volterra operator Tg acting on the spaces Ap α . This will be treated in greater detail in Chapter 6. See also [60]. We recall that, in the case of the Hp -spaces, the norm of the evaluations δs , s∈C1/2 , was computed in [ 15 ], where Bayart showed that ∥δs∥(Hp)∗ = ζ (2 σ ) 1 p . However, for the case of the spaces Ap α just some upper and lower estimates were known from [ 14 ]. The purpose of this chapter is to provide a proof of the estimate of the norm of these functionals. In order to estimate the norm of the evaluations, we will use a different approach to the one originally used in [ 14 ]. More precisely, we shall use the Riemann-Liouville operator, which contains the integration operator as a particular case. As we will see, the Riemann-Liouville operator It acts on Dirichlet series as a coefficient multiplier. This nature of the operator will 105 5. Evaluation functionals on the spaces Ap α provide a way of writing the Ap α -norm of a function f from the space in terms of the norm of Itf in some space Ap β , with β = β ( α, t ), α > β > − 1. A key ingredient to do so is a Cauchy integral type formula for the operator It . Section 5.2 is devoted to this operator and to the proof of all the associated results, as well as to establish this norms relationship between fand Itf. In [ 42 ], it was shown that the integration operator is bounded on Hp ∞ . In particular, given a Hp function vanishing at infinity, its primitive is in Hp too. In Section 5.3, we show that, in fact, one cannot expect the primitive to belong to any Hq space with q > p . Even more, it does not even belong to any Aq α with q > p , α > − 1. In this section, we characterise this membership of the primitive of an Hp function to the spaces Aq α in terms of the injection between Hpand Aq α+qt. In Section 5.4, we will prove the main result of the Chapter, showing how all the ingredients previously introduced fit together. Section 5.1, although containing some elementary original results, is mainly expository and it is devoted to introduce all the background material which will be needed in the subsequent sections. We also recall that we will say that a probability measure µ on (0 , + ∞ )is admissible if it is such that 0 ∈supp ( µ ), where supp ( µ )denotes the support of the measure µ. This means that, for every ε > 0,µ((0, ε)) >0. Along the chapter we will use the notation f ( x ) ≲g ( x )if there is some constant C > 0such that |f ( x ) | ≤ C|g ( x ) | for all (appropriate) x . If we have simultaneously that f(x)≲g(x)and that g(x)≲f(x), we write f≈g. 5.1 Introduction 5.1.1 State of the art In their work, Bailleul and Lefèvre proved the following upper estimates for the norm of δs . Let us point out that they deduced the following theorem from a more general version for the spaces Ap µ,∞ and Ap µ (see [ 14 , Theorem 1]). However, as this chapter is devoted to the spaces Ap α , we shall state the corresponding version of the result. Theorem 5.1.1. ([ 14 , Corollary 1]) Let p≥ 1and α > − 1. There exists a constant c=c(α, p)such that for every s=σ+it,σ > 1/2, ∥δs∥(Ap α,∞)∗≤c σ−1/2α+2 p and ∥δs∥(Ap α)∗≤cσ σ−1/2α+2 p·(5.1) Regarding the lower estimates. The following theorem was proven in [ 14 , Remarks (i)-(iii), pp. 25-26], see [ 14 , Remark (i)] for the case p > 1and [ 14 , Remark (iii)] for the case p= 1. Theorem 5.1.2. Let 1≤p < ∞and s=σ+it,σ > 1/2. Then, i) if α∈(−1,0): ∥δs∥(Ap α)∗≳1 σ−1/2α+2 p·(5.2) 106 5.1. Introduction ii) If α≥0 : ∥δs∥(Ap α)∗≳1 (σ−1/2)α+2 p(1 + |log(2 Re(s)−1)|)1 p· Our improvement allows us to extend the lower estimate (5.2) to the range α > − 1, thus, being able to get rid of the log term in the denominator. This is, we prove in Theorem 5.4.2 that, for all s=σ+it,σ > 1/2, ∥δs∥(Ap α)∗≈σ σ−1/2α+2 p, α > −1.(5.3) Regarding the spaces Ap α,∞ , for which in the work [ 14 ] only the first estimate in (5.1) is settled, we obtain, for a > 1/2, the following ∥δs∥(Ap α,∞)∗≈1 (σ−1/2)α+2 p for every σ∈(1/2, a).(5.4) These results also provide an affirmative answer to Question 2 from [ 55 ]. Let us point out that in the work [ 55 ] the authors consider the family of measures ναwith densities dνα(σ) = 2α−1 Γ(α−1)σα−2dσ, α > 1, σ > 0.(5.5) This is, they get rid of the exponential term in the definition of the measures µα . Observe that the measures να are not probability measures, but it is still possible to define in an identical manner the spaces Ap να . In fact, it is not difficult to prove that the space Ap να is equal to Ap µα−2,∞ . Moreover, in these spaces the corresponding norms happen to be equivalent (see Lemma 5.2.11), giving those norms the same spaces as a result of the a priori different completion processes. Because of this, all the results obtained in this chapter for the space Ap α,∞ are also valid for the family of measures considered in [ 55 ]. 5.1.2 Some auxiliary results In the next Theorem 5.1.4 we will deduce an analogue of (5.2) for the space Ap α,∞ . This estimate will be a key ingredient in the proof of the main result of the chapter. Before, proving it, let us establish the next auxiliary lemma comparing the norm of the evaluation functionals on the spaces Ap α and Ap α,∞ . Lemma 5.1.3. Let p≥1,α > −1and s∈C1 2. We have ∥δs∥(Ap α,∞)∗≤ ∥δs∥(Ap α)∗≤1+2∥δs∥(Ap α,∞)∗. Proof. The first inequality is clear by restriction since Ap α,∞⊂ Ap α. For the second one, take any f∈ Ap α and notice that f−a1∈ Ap α,∞ , where a1 is the constant Dirichlet coefficient. Since |a1|≤∥f∥A1 α≤ ∥f∥Ap α (see [ 14 , Theorem 9], where in that paper w1 = 1 since µ is a probability measure), we have |f(s)|≤|a1|+|f(s)−a1|≤∥f∥Ap α+∥δs∥(Ap α,∞)∗∥f−a1∥Ap α ≤1+2∥δs∥(Ap α,∞)∗∥f∥Ap α which gives the result. ■ 107 5. Evaluation functionals on the spaces Ap α Let us mention that the exact same proof works for the more general framework Ap µ . Hence, the lemma remains true if, instead of the measures µα , we consider an admissible measure. The following two results will be needed in the forthcoming sections. Theorem 5.1.4. Let p≥ 1, α∈ ( − 1 , 0) and a > 1 2· There exists a constant c=c(α, p, a)such that for every s=σ+it, with σ∈(1/2, a), it holds that ∥δs∥(Ap α)∗≥ ∥δs∥(Ap α,∞)∗≥c (σ−1/2)α+2 p· Proof. By Theorem 5.1.2 i), we know that ∥δs∥(Ap α)∗≥c (σ−1/2)α+2 p· This, together with Lemma 5.1.3, gives ∥δs∥(Ap α,∞)∗≥c 2(σ−1/2)α+2 p−1 2≥c 3(σ−1/2)α+2 p as soon as σis small enough, say less than σ0∈(1/2, a). When σ∈ ( σ0, a ), we have, by simply testing e 2 ( s )=2 −s (whose norm is less or equal than 1), ∥δs∥(Ap α,∞)∗≥2−σ≥2−a≥c′ (σ0−1/2)α+2 p≥c′ (σ−1/2)α+2 p for a suitable c′>0not depending on σ. This gives the conclusion. ■ Observe that we cannot expect a uniform polynomial lower estimate for ∥δs∥(Ap α,∞)∗ . Indeed we have that ∥δs∥(Ap α,∞)∗ = O 2 −σ when σ→ + ∞ . This is a consequence of both the fact that, for σ > 1/2, ∥δσ∥(Ap µ)∗≤Cµ∥δσ−1/2∥(Hp ∞)∗(5.6) and that ∥δβ∥(Hp ∞)∗≤C2−β, β ≥2.(5.7) Regarding (5.6) , let f∈ Ap µ,∞ . Then, by Lemma 2.5.5, we have that f1/2(·) = f(·+ 1/2) ∈ Hp ∞. Using this, we have that |f(σ)|=f1/2(σ−1/2) ≤ ∥δσ−1/2∥(Hp ∞)∗∥f1/2∥Hp ∞ ≤Cµ∥δσ−1/2∥(Hp ∞)∗∥f∥Ap µ,∞. For the estimate (5.7) , we take f∈ Hp ∞ with f ( s ) = Pn≥2ann−s and β≥ 2, so that |f(β)|=X n=2 ann−β≤X n≥2|an|n−β≤ ∥f∥Hp2−βX n≥22 nβ ≤ ∥f∥Hp2−βX n≥2 4 n2<∞, 108 5.2. The Dirichlet-Riemann-Liouville operator where in the second inequality we have used that for g ( s ) = Pn≥1ann−s and n∈Nfixed, the functional g7→ anis bounded on Hp(see [50, Remark 11.7]). We conclude this section with the following estimate of the H2 -norm of the m -powers, m∈N , of sufficiently large translates of the Riemann zeta function. This result will be needed in Section 5.3. Theorem 5.1.5. ([ 14 , Corollary 5]) Let m≥ 1be an integer and σ > 1 / 2. Then, there exists a positive constant c=c(m)such that ∥ζm(σ+·)∥2 H2≈c (2σ−1)m2,when σ→1 2 + . 5.2 The Dirichlet-Riemann-Liouville operator The idea behind our approach to estimate the norm of the functionals δs , s∈C1/2 , is the use of the Riemann-Liouville operator to write the norm of the Ap µfunctions. Definition 5.2.1. Let f be an analytic function in the right half-plane C+ , exponentially small when Res→ ∞ . For t > 0, we define the RiemannLiouville operator acting on fas It(f)(s) = 1 Γ(t)Z+∞ 0 xt−1f(x+s)dx, where Γ(t)stands for the Gamma function at the point t. Observe that It(f)is well defined in C+and it is holomorphic there. We are interested in studying the operator It acting on Dirichlet series. As we are about to see, for Dirichlet series to be stable under the action of the operator It , we must require their first coefficient to be zero. In particular, for t= 1, the operator Itcorresponds to the integration operator given by I(f)(s) = Z+∞ 0 f(u+s)du. See [ 42 ] for an exhaustive study of the boundedness of this operator on the Hardy spaces Hp. With this consideration, the operator It acts on Dirichlet series, up to the sign, as a fractional integration operator. Lemma 5.2.2. Let f ( s ) = P∞ n=2 ann−s be a Dirichlet series convergent in C+ . Then, It(f)(s) = ∞ X n=2 an (log n)tn−s, s ∈C+, t > 0.(5.8) 109 5. Evaluation functionals on the spaces Ap α Proof. We first prove the result for Dirichlet monomials. Indeed, given f(s) = n−s,n∈N,n≥2, we have that It(f)(s) = 1 Γ(t)Z+∞ 0 ut−1n−(s+u)du =n−s Γ(t)Z+∞ 0 ut−1e−ulog(n)du =n−s (log n)t· Now, let f be a Dirichlet series convergent in C+ and vanishing at infinity. By Lemma 2.1.4, we know that σa(f)≤1, hence ∞ X n=2 |an|n−σ<∞,for σ > 1. Taking this into account, we have that ∞ X n=2 Z+∞ 0|anxt−1n−x−s|dx = ∞ X n=2 Z+∞ 0|an|xt−1n−x−σdx = ∞ X n=2 |an| (log n)tn−σ<∞, so that we can apply Beppo Levi’s theorem to get It(f)(s) = ∞ X n=2 an (log n)tn−s, s ∈C1. In order to see that this representation actually holds in the whole right halfplane, it suffices, by Proposition 2.1.2, to check that the series converges at every point s0 in C+\C1 . Hence, let SNf ( s ) = PN n=1 ann−s , s∈C+ . We also set λn= (log n)−t. Then, by Abel’s summation formula SN ∞ X n=2 an (log n)tn−s!(s0) = N X n=2 anλnn−s0 = N X n=2 (Snf−Sn−1f)(s0)λn =SNf(s0)λN+ N−1 X n=2 Snf(s0)(λn−λn+1)· If we let N go to + ∞ , the first term goes to zero, since λN→ 0as N→ ∞ , and SNf ( s0 )is bounded. The second term is convergent, thanks to the fact that Snf ( s0 )is bounded and λn↘ 0. Observe that, in particular, we have shown that σc(It(f)) ≤σc(f)≤0.■ Definition 5.2.3. Let h be a measurable positive function in (0 ,∞ )such that ∥h∥L1(R+)= 1. i) We say that h satisfies the H -condition for p≥ 1if there exists a function q: (0,∞)→(0,∞)such that Z+∞ 0 xt−1q(1 x+1 )1/p (x+ 1)t+1 p dx < ∞,(5.9) 110 5.2. The Dirichlet-Riemann-Liouville operator and that for almost every λ∈(0,1), h(λu)≤q(λ)h(u),for a.e. uon (0,∞). ii) We say that hsatisfies the D-condition if there exists C > 0such that h(2u)≤Ch(u),for a.e. uon (0,∞). Example 5.2.4. The integrability hypothesis in the H -condition might look cumbersome. However, many natural choices for q satisfy (5.9) , even for all p . That is the case of the positive real functions on (0 ,∞ ), q ( λ ) = λα , α > − 1. For this choice of qthe integral in (5.9) becomes C(t) = Z+∞ 0 xt−1 (x+ 1)t+1 p+α p dx. This integral converges for any t > 0. Indeed, when x→ ∞, xt−1 (x+ 1)t+1 p+α p∼1 x1+ 1+α p and 1+(1+ α ) /p > 1. On the other hand, since t > 0, the term xt−1 guarantees the integrability in a neighbourhood of zero. The following fractional version of the classical Cauchy integral Theorem will be crucial in the proof of the forthcoming results of the chapter. Lemma 5.2.5. (i) Let kt=Z+∞ −∞ e1−iy (1 −iy)t+1 dy, t > 0. We have kt=2π Γ(t+ 1) = 0. (ii) Let fbe a Dirichlet polynomial. Then, f(θ) = 1 ktZ+∞ −∞ It(f)(iτ) (θ−iτ)t+1 dτ, θ > 0. Proof. First we focus on kt, which is clearly well-defined. For x∈R, let g(x) = 1 Γ(t+ 1)xte−ax1R+(x), a > 0. Obviously, g∈L1 ( R )and, for every y∈R , we compute its Fourier transform as bg(y) =: ZR g(x)e−ixydx. In order to obtain the expression of bg , we apply the Cauchy Theorem to the function z7→ F(z) = zte−az = exp−az +tlog z, holomorphic on C\R− and continuous at 0(when defining F (0) = 0), on a suitable path depending on y . More precisely, for y > 0and R > 0, consider the non-symmetric cone with vertex at zero given by Cy=nx+iu : 0 < x < R, 0< u < y axo. 111 5. Evaluation functionals on the spaces Ap α Indeed, since (i) holds and using Theorem 5.2.12, ∥ζm σ−1∥Hp≳∥It(ζm σ−1)∥Aq α≈ ∥ζm σ−1∥Aq α+qt ≥ ∥ζm σ∥Aq α+qt −1.(5.13) On the other hand, using that for all f∈ Hp , |a1|≤∥f∥Hp , together with the fact that the first coefficient of ζm σ is 1for every non-negative integer m and every σ > 1/2, we have that ∥ζm σ−1∥Hp≤ ∥ζm σ∥Hp+ 1 ≤2∥ζm σ∥Hp.(5.14) Putting together (5.13) and (5.14) gives the claim. Hence, using the claim and the definition of the norm Ap α, ∥ζm σ∥q Hp≳∥ζm σ∥q Aq α+qt =Z+∞ 0∥ζm σ+u∥q Hqdµα+qt(u) =Z+∞ 0∥ζmq/2 σ+u∥2 H2dµα+qt(u) ≥Z2σ−1 0∥ζmq/2 σ+u∥2 H2dµα+qt(u). Now, by Theorem 5.1.5 and since e −2u≈ 1on [0 , 2 σ− 1] when σ→ 1 / 2 + , we find that Z2σ−1 0∥ζmq/2 σ+u∥2 H2dµα+qt(u)≈Z2σ−1 0 1 (2σ+ 2u−1)m2q2 4 uα+qtdu ≥Z2σ−1 0 1 3(2σ−1)m2q2 4 uα+qtdu ≈1 (2σ−1)m2q2 4−α−qt−1· On the other hand, again by Theorem 5.1.5, we have that ∥ζm σ∥p Hp=∥ζ mp 2 σ∥2 H2≈1 (2σ−1)m2p2 4· Hence, still having in mind that σ→1/2+, we should have m2pq 4≥m2q2 4−(qt +1+α). That is, p≥q−4(qt +1+α)/qm2, for infinitely many non-negative integers m . However, this is false when m is large enough. So (i) implies (iii). ■ Remark 5.3.2.In particular, we have the following statement for the integration operator: Let 1≤p, q < ∞,α > −1. Then, the following assertions are equivalent: 1. I:Hp ∞→ Aq α,∞is bounded. 118 5.4. Main result 2. Id :Hp→ Aq α+qis bounded. 3. p≥q. As a consequence of a theorem due to Bayart, see [ 16 , p. 50], one has that the integration operator I does not map H1 into H2 (our statement here is stronger). In fact, using Bayart’s result and the coefficient estimates from [ 14 , Theorem 9] some cases of the equivalence between 1) and 3) can also be deduced. The equivalence between 3) and 1) was proven in [ 42 , Theorem 3.1] for p = q and from Hp ∞ into itself. In fact, what the case t = 1 of the previous theorem says is that given a Dirichlet series with vanishing first coefficient, in general, one cannot expect its primitive to belong to any Hq -space, q > p ; not even to any Aq α -space. On the other hand, the equivalence between 2) and 3) was established in [ 14 , Corollary 4] for the spaces Aq α with α = 0. Fu, Guo, and Yan using a different argument proved in [ 55 , Lemma 5.9] the compactness of the integration operator Ion Hpand on Ap αfor every p≥1and α > −1. 5.4 Main result We recall that the norm Hp remains invariant under vertical translations. In particular, this has as immediate consequence the vertical invariance of the norm Ap µ. This implies that ∥δu∥(Ap α)∗=∥δRe(u)∥(Ap α)∗for u∈C1/2. Lemma 5.4.1. The functions defined on (1 2,+∞)given by u7→ ∥δu∥(Ap µ)∗, u 7→ ∥δu∥(Ap µ,∞)∗ are non-increasing. Proof. Let u > v > 1/2. Then, by Theorem 2.2.23 b), |f(u)|=|fu−v(v)| ≤ ∥fu−v∥Ap µ∥δv∥(Ap µ)∗≤ ∥f∥Ap µ∥δv∥(Ap µ)∗. Therefore, ∥δu∥(Ap µ)∗≤ ∥δv∥(Ap µ)∗ , and the conclusion follows. The proof is identical for Ap µ,∞.■ We are now ready to establish the main result of this chapter. Theorem 5.4.2. Let p≥1,α > −1and a > 1 2.Then ∥δσ∥(Ap α,∞)∗≈1 (σ−1/2)α+2 p for every σ∈(1/2, a),(5.15) ∥δσ∥(Ap α)∗≈σ σ−1/2α+2 pfor every σ > 1 2,(5.16) In particular, ∥δσ∥(Ap α,∞)∗≈ ∥δσ∥(Ap α)∗≈1 (σ−1/2)α+2 p when σ→1 2 + ·(5.17) where the underlying constants depend on pand α(and on ain (5.15)) only. 119 5. Evaluation functionals on the spaces Ap α Proof. It suffices to prove the lower estimates since the upper ones are known from Theorem 5.1.1. Let us first show the lower estimate in (5.15). To this purpose, let P be a Dirichlet polynomial vanishing at infinity. Then, there exists a Dirichlet polynomial f , vanishing also at infinity, such that It ( f ) = P . Taking this into account and applying Theorem 5.2.12, we have that ∥f∥Ap α+tp ≈ ∥Itf∥Ap α=∥P∥Ap α,(5.18) where the constants depend on α,p, and tonly. Recall that thanks to Theorem 5.1.4, we already have the result when α∈ ( − 1 , 0). So, set α≥ 0and α0∈ ( − 1 , 0). Choose t > 0so that α = α0 + tp and fix a > 1 / 2. The boundedness of the functional δs , s∈C1/2 , on Ap α,∞ , α > 0, together with (5.18) yield |P(s)|=|It(f)(s)|=Z+∞ 0 ut−1 Γ(t)f(u+s)du ≤Z+∞ 0 ut−1 Γ(t)∥δu+s∥(Ap α0+tp,∞)∗∥f∥Ap α0+tp du ≈Z+∞ 0 ut−1 Γ(t)∥δu+s∥(Ap α,∞)∗∥P∥Ap α0du. Hence, taking the supremum over all Dirichlet polynomials P such that ∥P∥Ap α0= 1, we find that Γ(t)∥δs∥(Ap α0,∞)∗≲Z+∞ 0 ut−1∥δu+s∥(Ap α,∞)∗du. (5.19) Once more, by the invariance under vertical translations of the norm Ap µ , we can assume that s is real and suppose also that s<a . We pick some positive λ (we shall choose it large enough later). Let us now split the latter integral into two integrals, applying Lemma 5.4.1 to the first term and Theorem 5.1.1 to the second one so that, for some c > 0(coming from Th. 5.1.1), Zλ(s−1 2) 0 ut−1∥δu+s∥(Ap α,∞)∗du +Z+∞ λ(s−1 2) ut−1∥δu+s∥(Ap α,∞)∗du ≤ ∥δs∥(Ap α,∞)∗Zλ(s−1 2) 0 ut−1du +cZ+∞ λ(s−1 2) ut−1 (u+s−1/2)α+2 p du ≤λt t(s−1/2)t∥δs∥(Ap α,∞)∗+cZ+∞ λ(s−1 2) du uα0+2 p+1 du =λt t(s−1/2)t∥δs∥(Ap α,∞)∗+K(λ) (s−1/2)α0+2 p , where K ( λ ) = pc α0+2 λ−α0+2 p and K ( λ ) → 0as λ→ ∞ . Plugging this estimate in (5.19) and then using Theorem 5.1.4, we get c0Γ(t) (s−1/2)α0+2 p≤Γ(t)∥δs∥(Ap α0,∞)∗≲λt t(s−1/2)t∥δs∥(Ap α,∞)∗+K(λ) (s−1/2)α0+2 p· 120 5.4. Main result where c0>0and it depends only on α,p, and a. Now, by the definition of K ( λ ), we can choose λ large enough so that K(λ)<1 2c0Γ(t), giving 1 (s−1/2)α+2 p =1 (s−1/2)α0+2 p+t≲∥δs∥(Ap α,∞)∗. This proves (5.15) . Observe also that, thanks to Lemma 5.1.3, we clearly obtain (5.17). Finally, to prove the lower estimate in (5.16) , testing the constant functions, which we are not allowed for Ap α,∞ , we have that ∥δs∥(Ap α)∗≥ 1, for every s∈C1 2. Therefore, ∥δs∥(Ap α)∗≥max n1;∥δs∥(Ap α,∞)∗o. Now, if s=σ+it and σis, say, less or equal than 1, then, by (5.15) ∥δs∥(Ap α,∞)∗≳σ σ−1/2α+2 p· In case σ > 1, then 1≳σ σ−1/2α+2 p· Hence, ∥δs∥(Ap α)∗≥max n1;∥δs∥(Ap α,∞)∗o≳σ σ−1/2α+2 p·■ 121 CHAPTER 6 Volterra operators on Bergman spaces of Dirichlet series In previous chapters, we have dealt with different Banach spaces of Dirichlet series, like the spaces Hp or the algebra of Dirichlet series A ( C+ ). We have also considered different problems in these frameworks. As in the case of the classical Banach spaces of analytic functions in the unit disc, in the Dirichlet series setting, some of the most remarkable operators studied are the composition operators ([ 61 ], [ 15 ], [ 12 ], [ 13 ]), already treated in Chapter 3and Chapter 4, or the Volterra operators, which shall be treated in this chapter. For a convergent Dirichlet series g ( s ) = P∞ n=1 ann−s , the Volterra operator of symbol g , denoted by Tg , acting on a convergent Dirichlet series f is defined as Tgf(s) := −Z∞ s f(w)g′(w)dw, where the integral is computed along any curve from the point s to ∞ , approaching non-tangentially to ∞ . For instance, a curve Γ( t ) = s + t , t∈ [0 , + ∞ ). A first remarkable difference of the operator Tg on Banach spaces of Dirichlet series with respect to the unit disc setting is that the integration operator cannot be embedded in the study of the operator Tg . Indeed, in the unit disc context, considering as symbol g the identity map, Tg becomes the integration operator. Nonetheless, since the identity map is not a Dirichlet series, these operators must be studied separately when dealing with Dirichlet series. In fact, in Chapter 5a more detailed study of the integration and integration-type operators acting on the spaces Ap α was conducted. The reader is addressed to that chapter for further details. In Chapter 2, the Bergman type spaces Ap µ of Dirichlet series were defined. We recall that, in this context, µ stands for a probability measure on (0 , + ∞ ) such that 0 ∈supp ( µ ), this is, µ is an admissible measure. In Chapter 5, we chose a particular family of measures µα and considered a specific problem for the resulting spaces Ap α . In this chapter, we will work in a more general framework, namely, in a larger class of admissible measures. Being these the ambient spaces, the chapter is devoted to the study of the boundedness of the operator Tgacting on such spaces. The firsts to study the Volterra operator Tg on Banach spaces of Dirichlet series were Brevig, Perfekt, and Seip, in their work [ 39 ]. They considered 123 6. Volterra operators on Bergman spaces of Dirichlet series such operator acting on the spaces Hp . In such work, it was shown that the membership of the symbol g to the space BMOA ( C+ )is a sufficient condition in order to ensure the boundedness of Tg on Hp . Conversely, if the operator Tg is bounded on Hp and, surprisingly, p∈Q , p≥ 1, then the symbol g is a Dirichlet series belonging to the space BMOA ( C1/2 ). Furthermore, the previous sufficient condition fails to be necessary and so is the case for the necessary condition. To the best of the author’s knowledge since the appearance of this work, no improvements on these conditions have been made so far. However, seen the difficulty of achieving any enhancement, the research works which have appeared ever since have focused on studying the operators Tg on different Banach spaces of Dirichlet series. These spaces happen to be the Dirichlet series analogues to the classical Bergman spaces of the unit disc. The first space of such kind was introduced by McCarthy in 2004 and later embedded in a larger family of Bergman spaces of Dirichlet series by Bailleul and Lefèvre (see [ 14 ] for the original work and Chapter 2for a more detailed exposition). In their work, Bailleul and Lefèvre actually defined two new families of Bergman spaces of Dirichlet series: the spaces Ap µ , the ones studied in this thesis, and the Bp spaces. The family of spaces Bp is contained in the larger class Hp w , later introduced in the literature by Bommier-Hato [ 31 ]. The first two works on the Volterra operator Tg published after Brevig et al.’s paper ([ 31 ], [ 42 ]), studied this operator acting on the spaces Hp w. Given a symbol g depending on finitely many primes, this is, its Bohr lift depends on finitely many variables, Bommier-Hato [ 31 ] showed the sufficiency of the membership of g to the Bloch space Bloch ( C+ )to guarantee the boundedness of Tg on H2 w . In the same work, she also showed that the boundedness of Tg on H2 w necessarily implies the membership of the symbol g to the space Bloch ( C1/2 ) of Dirichlet series. The second work treating the topic, [ 42 ], contained some partial improvements, as the authors passed from the Hilbert case p = 2 to the range p∈ [1 ,∞ ), as well as removing the restriction on the Bohr lift of g . However, they restrained themselves to smaller classes of symbols. Indeed, Chen and Wang showed that the membership of the symbol g to the smaller space BMOA ( C+ )is sufficient for the boundedness of Tg on Hp w . On the other hand, for the case of 1-homogeneous symbols g, this is, when gis of the form g(s) = X pprime app−s, they prove that Tgis bounded on Hp wif and only if g∈ H2 w. Regarding the family of spaces Ap µ , the first work on the matter [ 55 ], signed by Fu, Guo, and Yan, is quite recent. There, the authors chose a specific family of measures να , α > 1, see (5.5) . However, as it was pointed out in Chapter 5, these measures are not admissible measures. Nonetheless, the resulting spaces coincide with the spaces Ap α−2,∞ . The authors proved that if the symbol g is a Dirichlet series belonging to the space Bloch ( C+ ), then the operator Tg acts boundedly on Ap να ,1 ≤p < ∞ . Fu et al. also showed that whenever Tg acts boundedly on the Hilbert space A2 να , the symbol g is a Dirichlet series belonging to the space Bloch(C1/2). 124 6.1. A family of Bloch spaces of Dirichlet series The sufficient condition and the necessary condition for the boundedness of Tg which we will provide in this chapter contain as a particular case the results obtained in [ 55 ]. The key point being that, thanks to Lemma 5.2.11, the spaces Ap να can be regarded as Ap α−2,∞ -spaces, with α > 1. Concerning the sufficiency, we show that if g is a Dirichlet series in the Bloch -type space Blochµ ( C+ ), then Tg is bounded on Ap µ ,1 ≤p < ∞ (see Theorem 6.4.1). In particular, when µ = να , α > 1, we have that Blochνα ( C+ ) = Bloch ( C+ ), for all α > 1. Moreover, we also show that ∥Tg∥L(Ap µ)≤C∥g∥Blochµ(C+), where C depends on p , but not on g . Concerning the necessity, we restrict ourselves to the the spaces Ap α. This is, we take µ=µα, where, we recall, dµα(σ) = 2α+1 Γ(α+ 1)σαe−2σdσ, α > −1. In this case, we are able to pass from the case p = 2 studied in [ 55 ], to the range 1 ≤p < ∞ . More precisely, in Section 6.4, we show that if Tg is bounded on Ap α , then g∈Bloch ( C1/2 ). A key ingredient to achieve this improvement is Theorem 5.4.2. The chapter is organised as follows. In Section 6.1 we present the new family of Bloch spaces of Dirichlet series Blochµ ( C+ ). This is a family of specific weighted Bloch spaces whose weight depends on the measure µ . We shall prove some of its most essential properties, such as boundedness abscissa, membership of vertical limits, inclusion relations, or estimates of the size of coefficients. We also give some examples of measures µ for which the resulting Blochµ space is a familiar weighted Bloch space. Section 6.2 is essentially expository and it is dedicated to recall all the background material from the Hp theory needed in the proof of the main results. In the same section, we will also present some Ap µ analogues of these results which will be useful later in the chapter. Section 6.3 contains the proof of a Littlewood-Paley identity for the Ap µ -spaces. These identities, as in the unit disc setting, happen to be essential when proving boundedness of Volterra operators on Banach spaces of analytic functions. Eventually, in the last section, Section 6.4, we prove the sufficient condition (see Theorem 6.4.1), which is based on the proof of a previous result on a Carleson measure type condition (see Theorem 6.4.5). In the second part of the section, the necessary condition is proven in Theorem 6.4.9. As in Chapter 5, we will use the notation f ( x ) ≲g ( x )if there is some constant C > 0such that |f ( x ) | ≤ C|g ( x ) | for all (appropriate) x . If we have simultaneously that f(x)≲g(x)and that g(x)≲f(x), we write f≈g. Throughout the chapter, in order to avoid some line overfulls, we will write ∞instead of +∞in the integral limits, even though the integrals are real. 6.1 A family of Bloch spaces of Dirichlet series The following Bloch spaces were introduced in [31]. 125 6. Volterra operators on Bergman spaces of Dirichlet series Definition 6.1.1. Let θ≥ 0. An analytic function f in the half-plane Cθ belongs to the Bloch space Bloch(Cθ)if sup σ+it∈Cθ (σ−θ)|f′(σ+it)|<∞.(6.1) These spaces are Banach spaces of analytic functions with the norm ∥f∥Bloch(Cθ):= sup σ+it∈Cθ (σ−θ)|f′(σ+it)|+|f(1)|. In particular, the space Bloch(Cθ)∩D is a closed subspace of Bloch(Cθ). Remark 6.1.2.Observe that, by Cauchy’s inequality, we have that H∞ ( C+ ), the space of bounded analytic functions in C+ , is a subspace of Bloch ( C+ ). In particular, H∞⊂Bloch(C+)∩D. Obviously, this is true for every half-plane Cθ,θ≥0. Remark 6.1.3.For θ≥ 0, we also have that Bloch ( Cθ ) ∩D ⊂ H∞ ( Cθ+ε ), for every ε > 0. Let us point out that this was already observed for the case θ = 0 in [ 31 , Lemma 3]. We prove it for θ = 0, being the argument identical for the general case. Let ε > 0. Since f∈ D , there exists M > 0such that f∈ H∞ ( CM ). If M≤ε , we are done. Hence, we suppose M > ε . Take s∈Cε , then |f(s)| ≤ |f(s+M−ε)|+Z[s,s+M−ε]|f′(z)||dz| ≤ ∥f∥H∞(CM)+ (M−ε)A ε<∞. Therefore, fis bounded in every half-plane Cε,ε > 0, as desired. Since the spaces were introduced in relation to the boundedness of the Volterra operator Tg on some Hilbert spaces of Dirichlet series, the most remarkable cases are θ = 0 and θ = 1 / 2. It is also worth mentioning that the space Bloch(C+)was treated with considerable detail in [67]. Definition 6.1.4. Let ω: (0,1] →(0,∞)be a measurable map such that ω∈L∞([ε, 1]) and 1/ω ∈L∞([ε, 1]),for all ε > 0.(6.2) We define the Blochω ( C+ )space as the collection of analytic functions f in C+ such that esssup 0<σ≤1 t∈R ω(σ)|f′(σ+it)|+∥f∥H∞(C1)<∞. Remark 6.1.5.The first condition in (6.2) ensures that the spaces Blochω ( C+ ) have an interesting structure since, otherwise, the spaces merely consist of constant functions. Remark 6.1.6.The second condition in (6.2) guarantees that the norm convergence implies local uniform convergence. This ensures that these spaces Blochω(C+)are Banach spaces of analytic functions with the norm ∥f∥Blochω(C+):= ess sup 0<σ≤1 t∈R ω(σ)|f′(σ+it)|+∥f∥H∞(C1). Clearly, we also have that Blochω ( C+ ) ∩D is a closed subspace of Blochω ( C+ ). 126 6.1. A family of Bloch spaces of Dirichlet series Remark 6.1.7.In the case ω(σ) = σ, we have that Blochω(C+)⊊Bloch(C+). Let us see this. Consider f∈Blochω(C+). Then, sup 0<σ≤1 t∈R σ|f′(σ+it)|+∥f∥H∞(C1)<∞. Hence, by definition, f∈H∞ ( C1 ). Observe also that, by the same argument from Remark 6.1.3, f∈H∞ ( C1/2 ). Then, by Cauchy’s inequality, we have that (σ−1/2)|f′(σ+it)| ≤ M, for all σ > 1/2and t∈R. Then, σ 2|f′(σ+it)| ≤ M, for all σ > 1and t∈R. Putting all together, we obtain sup σ>0 t∈R σ|f′(σ+it)|<∞, as desired. For the strict inclusion, it suffices to consider the function f(z) = log z,z∈C+. Clearly, σ|f′(σ+it)| ≤ 1. However, fis not bounded in any half-plane Cε,ε > 0. Remark 6.1.8.For the special choice ω(σ) = σ, we have that Blochω(C+)∩D =Bloch(C+)∩D. The fact that Blochω ( C+ ) ∩D ⊂ Bloch ( C+ ) ∩D follows immediately from the previous remark. For the converse inclusion, let f∈Bloch ( C+ ) ∩D . Remark 6.1.3 gives the membership of f to H∞ ( C1 ). Regarding the first term of the norm Blochω(C+), it is clearly finite since sup 0<σ≤1 t∈R σ|f′(σ+it)| ≤ sup 0<σ t∈R σ|f′(σ+it)|. Remark 6.1.9.Working as in Remark 6.1.3, we have that whenever 1 /ω is in L1 (0 , 1), the resulting space Blochω ( C+ )is contained in H∞ ( C+ ), even in A ( C+ ), the algebra of bounded analytic functions in C+ which are also uniformly continuous in the same half-plane. Since we are interested in the Blochω spaces because of their connection with the operator Tg acting on the spaces Ap µ , we shall pay special attention to a family of Bloch spaces where the expression of the weight ω is intimately connected to the measure µ. Let h be a positive function such that Rε 0h ( σ ) dσ > 0for all ε > 0and h∈L1 (0 ,∞ ). In [ 14 ], Bailleul and Lefèvre introduced the following function associated to this density βh(σ) := Zσ 0 (σ−u)h(u)du. 127