Composition operators on Hardy-Orlicz spaces

arXiv:math/0610905v1 [math.FA] 29 Oct 2006 Comp osition op erators on Hardy-Orliz spaes Pasal Lefèvre, Daniel Li, Hervé Queéle, Luis Ro dríguez-Piazza 2nd February 2008 Abstrat. We investigate omposition operators on Hardy-Orliz spaes when the Orliz funtion Ψ grows rapid ly: ompatness, weak ompatness, to be p - summing, order bounded, . . . , and show how these notions behave aording to the growth of Ψ . We introdue an adapted version of Carleson measure. We onstrut various examples showing that our results are essential ly sharp. In the last part, we study the ase of Bergman-Orliz spaes. Mathematis Sub jet Classiation. Primary: 47 B 33  46 E 30; Se- ondary: Key-words. Bergman-Orliz spae  Carleson measure  omposition operator  Hardy-Orliz spae 1 Intro dution. Comp osition op erators on the lassial Hardy spaes Hp have b een widely studied (see [36℄, and [11℄, and referenes therein; see also [19℄ and [20℄, and [7℄, [10℄, [18℄, [28℄, [33℄, [37℄, [38℄ for some more reent works), but it seems that one has not paid muh attention to the Hardy-Orliz spaes (in [40℄ and [41℄, J.-O. Strömb erg studied Hardy-Orliz spaes in the ase when the Orliz funtion Ψ inreases smo othly; see also [26℄ for omp osition op erators). We shall investigate what happ ens when the Orliz funtion grows more rapidly than a p ower funtion. Reall that, given an analyti self-map φ:D→D of the unit disk D , the omposition operator asso iated to φ is the map Cφ:f7→ f◦φ . This map may op erate on various Banah spaes X of analyti funtions on D (Hardy spaes, Bergman spaes, . . . , and their weighted versions (see [42℄ for instane), Blo h spaes B and B0 , BMOA and V MOA , Dirihlet spaes (see [1℄), or some more general spaes as Nevanlinna or Smirnov lasses: [8℄, [9℄, [17℄, [21℄; see also [3℄, [4℄, [14℄ for omp osition op erators on Hp spaes of Dirihlet series, though they are not indued by an analyti self-map of D ). The goal is to link prop erties of the omp osition op erator Cφ:X→X (ompatness, strong or weak, for example) to prop erties of the symbol φ (essentially its b ehaviour near the frontier of D ). For that study, one an roughly sp eaking (see [11℄, Chapter 4, though their notions are dierent from ours) distinguish two kind of spaes. 1 1) The smal l spaes X ; those spaes are in a sense lose to the Hardy spae H∞ : the ompatness of Cφ:X→X is very restritive and it imp oses severe restritions on φ . For example, if X=H∞ , a theorem of J. Shwartz ([34℄) implies that Cφ:H∞→H∞ is ompat if and only if kφk∞<1 (weakly ompat sues: see [23℄), whih in turn implies reinfored ompatness properties for Cφ . For example, Cφ:H∞→H∞ is nulear and 1 -summing as soon as it is ompat. 2) The large spaes X ; those spaes are in a sense lose to the Hardy spae H1 : the ompatness of Cφ:X→X an take plae fairly often, and in general implies no self-improvement. For example, for X=H2 , Cφ:H2→H2 an b e ompat without b eing Hilb ert-Shmidt, even if φ is injetive ([39℄, Theorem 6.2). Another formulation (whih lends b etter to generalizations in the non- Hilb ertian ase) is that Cφ an b e non-order b ounded (see [16℄, and our Setion 3) and yet ompat. In this pap er, we shall rather b e on the small spae side, sine we shall work in spaes asso iated to a very large Orliz funtion Ψ (typially: Ψ(x) = Ψ2(x) = ex2−1 ), and the previous situation will not take plae: our op erators will b e e.g. order-b ounded as so on as they are (weakly) ompat, even if the situation is not so extreme as for H∞ . However, for slightly smaller Orliz funtions (for instane Ψ(x) = exp log(x+ 1)2−1 ), the situation is loser to the H2 ase: the omp osition operators may be ompat on HΨ , but not order-b ounded (Theorem 4.22). This pap er is divided into ve parts. Setion 1 is this Intro dution. In Se- tion 2, whih is essentially notational, we reall some more or less standard fats on Orliz funtions Ψ , on asso iated Orliz spaes LΨ , and the little Orliz spae MΨ , and their banahi properties, asso iated with various slow growth onditions (indiated by subsripts: ∆1 , ∆2 , . . . ) or fast growth onditions (indiated by sup ersripts: ∆0 , ∆1 , ∆2 , . . . ). In Setion 3, we intro due the Hardy-Orliz spae HΨ , and its (or his) little brother HMΨ . These spaes have already been studied (see [15℄, [31℄), but rather for slowly growing funtions Ψ (having ∆2 most of the time), and their denition is not so learly outlined, so we give a detailed exp osition of the equivalene of the two natural denitions that one has in mind (if one wants to extend the ase of Hardy spaes Hp asso iated to Ψ(x) = xp ), as well as of the automati b oundedness (through the Littlewoo d subordination priniple and the ase of inner self-maps of the disk) of omp osition op erators on those spaes. Two of the main theorems are Theorem 3.24 and Theorem 3.27. Roughly sp eaking, Theorem 3.24 says the following: if Ψ is very fast growing (having ∆2 more preisely), HΨ is a small spae, the (weak) ompatness of Cφ is very restritive, and even if the situation is not so extreme as for H∞ ( kφk∞<1 ), φ has to tend to the boundary very slowly, and Cφ is automatially order-bounded into MΨ . However, Theorem 3.27 shows the limits of this self-improvement: Cφ may b e order-b ounded into MΨ (and hene ompat), but p -summing for no nite p . We also show that, when Ψ has ∆2 growth, there are always symb ols 2 φ induing ompat omp osition op erators on H2 (even Hilb ert-Shmidt), but not ompat on HΨ . Setion 4 is devoted to the use of Carleson measures. The usefulness of those measures in the study of omp osition op erators is well-known (see [12℄, [11℄, [6℄) and, to our knowledge, rst expliitly used for ompatness in [27℄. In partiular, we reall the following neessary and suient ondition for Cφ:H2→H2 to b e ompat: if h > 0 and w∈∂D , onsider the Carleson window : W(w, h) = {z∈D;|z| ≥ 1−h and |arg(z¯w)| ≤ h}. If φ is an analyti self-map of D with b oundary values φ∗ , and µφ=φ∗(m) denotes the image under φ∗ of the normalized Leb esgue measure (Haar measure) on T=∂D , the measure µφ is always a Carleson measure, i.e. : sup w∈∂D µφW(w, h)=O(h). Now, we an state: Theorem 1.1 (B. MaCluer [27℄) The omposition operator Cφ:H2→H2 is ompat if and only if µφ satises the  little-oh  ondition, i.e. if and only if: (1.1) sup w∈∂D µφW(w, h)=o(h) as h→0. There is another famous neessary and suient ompatness ondition, due to J. Shapiro ([35℄): Let us denote by Nφ the Nevanlinna ounting funtion of φ , i.e. : Nφ(w) = (P φ(z)=w log 1 |z| if w6=φ(0) and w∈φ(D) 0 if w /∈φ(D). By Littlewoo d's inequality, one always has (see [11℄, page 33): Nφ(w) = O(1 −|w|). Now, Shapiro's Theorem reads: Theorem 1.2 (J. Shapiro [35℄) The omposition operator Cφ:H2→H2 is ompat if and only if Nφ satises the  little-oh  ondition, i.e.: (1.2) Nφ(w) = o(1 −|w|) as |w|< →1. Theorem 1.2 is very elegant, and probably more popular than Theorem 1.1. Yet, it is diult to apply beause the assumption (1.2) is diult to hek. Here, we shall app eal to Theorem 1.1 to prove (Theorem 4.1) that the ompatness of Cφ:H2→H2 annot b e read on |φ∗| when φ is not nitely valent; more preisely, there are two analyti self-maps φ1 and φ2:D→D 3 suh that: |φ∗ 1|=|φ∗ 2|m - a.e. , but Cφ1:H2→H2 is not ompat, though Cφ2:H2→H2 is ompat. We show then that every omposition op erator whih is ompat on HΨ is neessarily ompat on Hp for all p < ∞ . However, there exist (see ab ove, or Setion 3) symbols φ induing ompat omp osition op erators on H2 but whih are not ompat on HΨ , when Ψ has ∆2 growth. Hene ondition (1.1) do es not sue to haraterize the ompat omposition op erators on HΨ . We have to replae Carleson measures and ondition (1.1) by what we may all  Ψ - Carleson measures, and an adaptated  little-oh  ondition, whih allows us to haraterize ompatness for omp osition operators. It follows that if Ψ∈∆0 , then the weak ompatness of Cφ:HΨ→HΨ implies its ompatness. We also show that the ab ove example φ2 indues, for an Orliz funtion Ψ whih does not satisfy ∆2 , but whih satises ∆1 , a omp osition op erator on HΨ whih is ompat, but not order b ounded into MΨ(T) (Theorem 4.22), showing that the assumption that Ψ∈∆2 in Theorem 3.24 is not only a tehnial assumption. In Setion 5, we intro due the Bergman-Orliz spaes. Let us remind that, in the Hilb ertian ase, the study of ompatness of omp osition op erators is simpler for the Bergman spae B2 than for the Hardy spae H2 . For example, we have the following: Theorem 1.3 (see [36℄) i) Cφ:B2→B2 is ompat if and only if (1.3) lim |z|< →1 1−|φ(z)| 1−|z|= +∞. ii) (1.3) is always neessary for Cφ:H2→H2 to be ompat, and it is suient when φ is injetive, or only nitely valent. iii) There are Blashke produts φ satisfying (1.3) for whih Cφ:H2→H2 is (in an obvious manner) non-ompat. We p erform here a similar study for the Bergman-Orliz spae BΨ , and ompare the situation with that of the Hardy-Orliz spae HΨ . We are naturally led to a reinforement of (1.3) under the form: (1.4) Ψ−11 (1 −|φ(a)|)2 Ψ−11 (1 −|a|)2−→ |a|< →1 0 (always neessary, and suient when Ψ is ∆2 ), whih reads, in the ase Ψ(x) = Ψ2(x) = ex2−1 : (1.5) 1−|φ(z)| ≥ cε(1 −|z|)ε for all ε > 0. 4 In [36℄, the onstrution of a Blashke pro dut satisfying (1.3) is fairly deliate, and app eals to Frostman's Lemma and Julia-Caratheo dory's Theorem on nonangular derivatives at the boundary. Here, we an no longer use these tools for the reinforement (1.5), so we do make a diret onstrution, using the Parseval formula for nite groups. In passing, the onstrution gives a simpler proof for (1.3). Otherwise, the theorem whih we obain is similar to Shapiro's one, if one ignores some tehnial diulties due to the non-separability of BΨ : we have to transit by the smaller Bergman-Morse-Transue spae BMΨ , whih is the losure of H∞ in BΨ , is separable, and has BΨ as its bidual. 2 Notation Let D b e the op en unit disk of the omplex plane, that is the set of omplex numbers with modulus stritly less than 1 , and T the unit irle, i.e. the set of omplex numb ers with mo dulus 1 . We shall onsider in this pap er Orliz spaes dened on a probability spae (Ω,P) , whih will be the unit irle T , with its (normalized) Haar measure m (most often identied with the normalized Leb esgue measure dx/2π on the interval [0,2π] ), or the op en unit disk D , provided with the normalized area measure A . By an Orliz funtion, we shall understand that Ψ: [0,∞]→[0,∞] is a non-dereasing onvex funtion suh that Ψ(0) = 0 and Ψ(∞) = ∞ . To avoid pathologies, we shall assume that we work with an Orliz funtion Ψ having the following additional prop erties: Ψ is ontinuous at 0 , stritly onvex (hene inreasing), and suh that Ψ(x) x−→ x→∞∞. This is essentially to exlude the ase of Ψ(x) = ax . If Ψ′ is the left (or instead, if one prefers, the right) derivative of Ψ , one has Ψ(x) = Rx 0Ψ′(t)dt for every x > 0 . The Orliz spae LΨ(Ω) is the spae of all (equivalene lasses of ) measurable funtions f: Ω →C for whih there is a onstant C > 0 suh that ZΩ Ψ|f(t)| CdP(t)<+∞ and then kfkΨ (the Luxemburg norm ) is the inmum of all p ossible onstants C suh that this integral is ≤1 . The Morse-Transue spae MΨ(Ω) is the subspae generated by L∞(Ω) , or, equivalently, the subspae of all funtions f for whih the above integral is nite for all C > 0 . To every Orliz funtion is asso iated the omplementary Orliz funtion Φ = Ψ∗: [0,∞]→[0,∞] dened by: Φ(x) = sup y≥0xy −Ψ(y), 5 The extra assumptions on Ψ ensure that Φ is itself stritly onvex. When Φ satises the ∆2 ondition (see the denition b elow), LΨ is (isomorphially, if LΦ is itself normed by the Luxemburg norm) the dual spae of LΦ , whih is, in turn, the dual of MΨ . 2.1 Growth onditions We shall have to use various growth onditions for the Orliz funtion Ψ . These onditions are usually denoted as ∆ -onditions. Our interest is in Orliz funtions whih have a somewhat fast growth. Usually, some of these onditions are dened through a mo derate growth ondition on the omplementary funtion Φ of Ψ , and the ondition ∆ for the Orliz funtion is translated as a ∇ -ondition for the omplementary funtion. So we shall distinguish b etween moderate growth onditions, that we shall dene for the omplementary Orliz funtion, and fast growth onditions. To emphasize this distintion, we shall denote, sometimes in hanging the usual notation (see [22, 30℄), the mo derate growth onditions with a subsript, and the fast growth onditions with a sup ersript. Mo derate growth onditions • The Orliz funtion Φ saties the ∆1 -ondition ( Φ∈∆1 ) if, for some onstant c > 0 , one has: Φ(xy)≤cΦ(x)Φ(y) for x, y large enough. This is equivalent to say that Φ(axy)≤Φ(x)Φ(y) for some onstant a > 0 and x, y large enough. This ondition is usually denoted by ∆′ (see [30℄, page 28). •Φ satises the ∆2 -ondition ( Φ∈∆2 ) if Φ(2x)≤KΦ(x) for some onstant K > 1 and x large enough. One has: Φ∈∆1⇒Φ∈∆2. 6 Fast growth onditions • The Orliz funtion Ψ satises the ∆0 -ondition ( Ψ∈∆0 ) if (see [25℄), for some β > 1 : lim x→+∞ Ψ(βx) Ψ(x)= +∞. A typial example is Ψ(x) = exp log(x+ 2) log log(x+ 2)−2log log 2 ; another is Ψ(x) = exp log(x+ 1)3/2−1 . • The Orliz funtion Ψ satises the ∆1 -ondition ( Ψ∈∆1 ) if there is some β > 1 suh that: xΨ(x)≤Ψ(βx) for x large enough. Note that this latter ondition is usually written as ∆3 -ondition, with a subsript (see [30℄, 2.5). This notation ts b etter with our onvention, and the sup ersript 1 agrees with the fat that this ∆1 -ondition is b etween the ∆0 - ondition and the following ∆2 -ondition. Ψ∈∆1 implies that Ψ(x)≥exp α(log x)2 for some α > 0 and x large enough (see [30℄, Prop osition 2, page 37). A typial example is Ψ(x) = e(log(x+1))2−1 . • The Orliz funtion Ψ: [0,∞)→[0,∞) is said to satisfy the ∆2 -ondition ( Ψ∈∆2 ) if there exists some α > 1 suh that: Ψ(x)2≤Ψ(αx) for x large enough. This implies that Ψ(x)≥exp(xα) for some α > 0 and x large enough ([30℄, Prop osition 6, page 40). A typial example is Ψ(x) = Ψ2(x) = ex2−1 . Conditions of regularity • The Orliz funtion Ψ satises the ∇2 -ondition ( Ψ∈ ∇2 ) if its omplementary funtion Φ satises the ∆2 -ondition. This is equivalent to say that for some onstant β > 1 and some x0>0 , one has Ψ(βx)≥2βΨ(x) for x≥x0 , and that implies that Ψ(x) x−→ x→∞∞ . In partiular, this exludes the ase LΨ=L1 . 7 • The Orliz funtion Ψ satises the ∇1 -ondition ( Ψ∈ ∇1 ) if its omplementary funtion Φ satises the ∆1 -ondition. This is equivalent to say that Ψ(x)Ψ(y)≤Ψ(bxy) for some onstant b > 0 and x, y large enough. All p ower funtions Ψ(x) = xp satisfy ∇1 , but Ψ(x) = xplog(x+ 1) do es not. One has (see [30℄, page 43): Ψ∈∆1+3Ψ∈∆0 & F F F F F F F F F F F F Ψ∈∆2 7? x x x x x x x x x x x x x x %- S S S S S S S S S S S S S S S S S S S S S SΨ∈ ∇2 Ψ∈ ∇1 2: l l l l l l l l l l l l l l l l l l l l But ∆1 do es not imply ∇1 . That ∇1 do es not even imply ∆0 is lear sine any power funtion Ψ(x) = xp ( p≥1 ) is in ∇1 . 2.2 Some sp ei funtions In this pap er, we shall make a repeated use of the following funtions: • If Ψ is an Orliz funtion, we set, for every K > 0 : (2.1) χK(x) = ΨKΨ−1(x), x > 0. For example, if Ψ(x) = ex−1 , then Ψ−1(x) = log(1 + x) , and χK(x) = (1 + x)K−1 . Note that:  Ψ∈∆0 means that χβ(u) u−→ u→∞+∞ , for some β > 1 .  Ψ∈∆1 means that χβ(u)≥uΨ−1(u) for u large enough, for some β > 1 .  Ψ∈∆2 means that χα(u)≥u2 for u large enough, for some α > 1 .  Ψ∈ ∇1 means that χA(u)≥(Ψ(A)/b)u for u large enough and for every A large enough, for some b > 0 . • For |a|= 1 and 0≤r < 1 , ua,r is the funtion dened on the unit disk D by: (2.2) ua,r(z) = 1−r 1−¯arz 2,|z|<1. Note that kua,rk∞= 1 and kua,rk1≤1−r . 8 3 Comp osition op erators on Hardy-Orliz spaes 3.1 Hardy-Orliz spaes It is well-known that the lassial Hp spaes ( 1≤p≤ ∞ ) an b e dened in two equivalent ways: 1) Hp is the spae of analyti funtions f:D→C for whih, setting fr(t) = f(reit) : kfkHp= sup 0≤r<1kfrkp is nite (reall that the numbers kfrkp inrease with r ). When f∈Hp , the Fatou-Riesz Theorem asserts that the b oundary limits f∗(t) = limr< →1fr(t) exist almost everywhere and kfkHp=kf∗kp . One has f∗∈Lp([0,2π]) , and its Fourier o eients b f∗(n) vanish for n < 0 . 2) Conversely, for every funtion g∈Lp([0,2π]) whose Fourier o eients ˆg(n) vanish for n < 0 , the analyti extension P[g]: D→C dened by P[g](z) = Pn≥0ˆg(n)zn is in Hp and g is the b oundary limit (P[g])∗ of P[g] . Hardy-Orliz spaes HΨ are dened in a similar way. However, we did not nd very satisfatory referenes, and, though the reasonings are essentially the same as in the lassial ase, the lak of homogeneity of Ψ and the presene of the two spaes MΨ and LΨ gives pro ofs whih are not so obvious and therefore we shall give some details. It should b e noted that our denition is not exatly the same as the one given in [30℄,  9.1. We shall b egin with the following prop osition. Prop osition 3.1 Let f:D→C be an analyti funtion. For every Orliz funtion Ψ , the fol lowing assertions are equivalent: 1) sup0≤r<1kfrkΨ<+∞ , where fr(t) = f(reit) ; 2) there exists f∗∈LΨ([0,2π]) suh that b f∗(n) = 0 for n < 0 and for whih f(z) = Pn≥0b f∗(n)zn , z∈D . When these onditions are satised, one has kf∗kΨ= sup0≤r<1kfrkΨ . Let us note that, sine Ψ is onvex and inreasing, Ψ(a|f|) is subharmoni on D , and hene the numb ers RTΨ(a|fr|)dm inrease with r , for every a > 0 . This proposition leads to the following denition. Denition 3.2 Given an Orliz funtion Ψ , the Hardy-Orliz spae HΨ asso- iated to Ψ is the spae of analyti funtions f:D→C suh that one of the equivalent onditions of the above proposition is satised. The norm of f is de- ned by kfkHΨ=kf∗kΨ . We shal l denote by HMΨ the Hardy-Morse-Transue spae , i.e. the subspae {f∈HΨ;f∗∈MΨ(T)} . In the sequel, we shall make no distintion between f and f∗ , unless there may b e some ambiguity, and shall write f instead of f∗ for the b oundary limit. 9 2) One has: kCφk ≥ 1 8Ψ−1(1)Ψ−11 + |φ(0)| 1−|φ(0)|. 3) When Ψ∈ ∇1 global ly: Ψ(x)Ψ(y)≤Ψ(bxy) for al l x, y ≥0 , we also have: kCφk ≤ bΨ−11 + |φ(0)| 1−|φ(0)|. 4) Moreover, Cφ maps HMΨ into HMΨ . Hene, if Ψ∈ ∇2 , then Cφ:HΨ→ HΨ is the bi-adjoint of the omposition operator Cφ:HMΨ→HMΨ . Note that when Ψ(x) = xp for 1≤p < ∞ , then Ψ∈ ∇1 globally, with b= 1 . Pro of. 1) Assume rst that φ(0) = 0 . Let f∈HΨ , with kfkΨ= 1 . Sine Ψ is onvex and inreasing, the funtion u= Ψ ◦ |f| is subharmoni on D , thanks to Jensen's inequality. The ondition φ(0) = 0 allows to use Littlewo o d's sub ordination priniple ([12℄, Theorem 1.7); for r < 1 , one has: Z2π 0 Ψ|(f◦φ)(reit)|dt 2π≤Z2π 0 Ψ|f(reit)|dt 2π≤1. Hene f◦φ∈HΨ and kf◦φkΨ≤1 . Assume now that φ is an inner funtion, and let a=φ(0) . It is known that (see [29℄, Theorem 1) that φ∗(m) = Pa.m , where φ∗(m) is the image under φ∗ (the b oundary limit of φ ) of the normalized Leb esgue measure m , and Pa.m is the measure of density Pa , the Poisson kernel at a . Therefore, for every f∈HΨ with kfkΨ= 1 , one has for 0≤r < 1 , in setting Ka=kPak∞=1+|a| 1−|a| : Z2π 0 Ψ|(f◦φ)(reit)| Kadt 2π≤ZT Ψ|(f◦φ)∗| Kadm =ZT Ψ|f∗◦φ∗| Kadm (reall that |φ∗|= 1 ) =ZT Ψ|f∗| Kadφ∗(m) = ZT Ψ|f∗| KaPadm (3.1) ≤ZT 1 Ka Ψ(|f∗|)Padm , sine Ka>1 ≤ZT 1 Ka Ψ(|f∗|)kPak∞dm =ZT Ψ(|f∗|)dm ≤1. Hene k(f◦φ)rkΨ≤Ka , and therefore kf◦φkΨ≤Ka . 16 Then, for an arbitrary φ , let a=φ(0) again, and let φa b e the automorphism z7→ z−a 1−¯az , whose inverse is φ−a . Sine φ=φ−a◦(φa◦φ) , one has Cφ= Cφa◦φ◦Cφ−a . But φ−a is inner and, on the other hand, (φa◦φ)(0) = 0 ; hene parts a) and b) of the pro of give: kCφk ≤ kCφ−ak ≤ Ka=1 + |a| 1−|a| , whih gives the rst part of the proof. 2) By Lemma 3.11, we have for every f∈HΨ with kfkΨ≤1 : |(f◦φ)(0)| ≤ kδ0k(HΨ)∗kf◦φkΨ≤2Ψ−1(1) kCφk. In other words: |fφ(0)| ≤ 2Ψ−1(1)kCφk for every suh f∈HΨ . Hene: kδφ(0)k(HΨ)∗≤2Ψ−1(1) kCφk, giving kCφk ≥ 1 8Ψ−1(1)Ψ−11 + |φ(0)| 1−|φ(0)|, by using Lemma 3.11 again, but the minoration. 3) When Ψ∈ ∇1 globally, we go bak to the pro of of 1) . We have only to mo dify inequalities (3.1) in b). Setting K′ a= Ψ−1(Ka) , and writing Pa= ΨΨ−1(Pa) , we get, for every f∈HΨ with kfkΨ= 1 : Z2π 0 Ψ|(f◦φ)(reit)| bK′ adt 2π≤ZT Ψ|f∗| bK′ aPadm =ZT Ψ|f∗| bK′ aΨΨ−1(Pa)dm ≤ZT Ψ|f∗| K′ a Ψ−1(Pa)dm ≤ZT Ψ(|f∗|)dm ≤1, sine Ψ−1(Pa)≤Ψ−1(kPak∞) = K′ a , giving kf◦φkΨ≤bK′ a . 4) Supp ose now that f∈HMΨ . As b efore, when φ(0) = 0 , Littlewo o d's sub ordination priniple gives, for every C > 0 : Z2π 0 ΨC|(f◦φ)(eit)|dt 2π= sup r<1Z2π 0 ΨC|(f◦φ)(reit)|dt 2π ≤sup r<1Z2π 0 ΨC|f(reit)|dt 2π=Z2π 0 ΨC|f(eit)|dt 2π<+∞; 17 hene f◦φ∈HMΨ . When φ is inner, the same omputations as in 1) b) ab ove, using that φ∗(m) = Pa.m , where a=φ(0) , give, for every C > 0 : Z2π 0 ΨC|(f◦φ)(reit)|dt 2π≤ZT Ψ(C|f∗|)kPak∞dm =KaZT Ψ(C|f∗|)dm < +∞, and f◦φ∈HMΨ again. The general ase follows, as in 1) ) ab ove, sine f∈HMΨ implies f◦φ−a∈HMΨ , b eause φ−a is inner, and then f◦φ= (f◦φ−a)◦(φa◦φ)∈HMΨ sine (φa◦φ)(0) = 0 .  3.4 Order b ounded omp osition op erators Reall that an op erator T:X→Z from a Banah spae X into a Banah subspae Z of a Banah lattie Y is order bounded if there is some p ositive y∈Y suh that |T x| ≤ y for every x in the unit ball of X . Before studying order b ounded omp osition op erators, we shall reall the following, ertainly well-known, result, whih says that order boundedness an b e seen as stronger than ompatness. Prop osition 3.13 Let T:L2(µ)→L2(µ) be a ontinuous linear operator. Then T is order bounded if and only if it is a Hilbert-Shmidt operator. The pro of is straightforward: if B is the unit ball of L2(µ) , and (ei)i is an orthonormal basis, one has supf∈B|T f|=Pi|T ei|21/2 . Hene supf∈B|T f| ∈ L2(µ) if and only if RPi|T ei|2dµ =PikT eik2<+∞ , i.e. if and only if T is Hilb ert-Shmidt. J. H. Shapiro and P. D. Taylor proved in [39℄ that there exist omposition op erators on H2 whih are ompat but not Hilb ert-Shmidt. We are going to see that, when the Orliz funtion Ψ grows fast enough, the ompatness of omp osition op erators on HΨ is equivalent to their order b oundedness. Prop osition 3.14 The omposition operator Cφ:HΨ→HΨ is order bounded (resp. order bounded into MΨ(T) ) if and only if Ψ−11 1−|φ|∈LΨ(T) (resp. Ψ−11 1−|φ|∈MΨ(T) ). Equivalently, if and only if (OB1) χA1 1−|φ|∈L1(T) for some A > 0, respetively: (OB2) χA1 1−|φ|∈L1(T) for every A > 0, 18 In other words (reall that χA(x) = ΨAΨ−1(x) ), if and only if: ZT ΨAΨ−11 1−|φ|dm < +∞ for some (resp. every) A > 0 . Remark For a sequene (gn)n of elements of LΨ(Ω) , one has kgnkΨ−→ n→+∞0 if and only if (3.2) ZΩ Ψ|gn| εdP−→ n→+∞0 for every ε > 0. In fat, if (3.2) holds, then for n≥nε , the ab ove integrals are ≤1 , and hene kgnkΨ≤ε . Conversely, assume that kgnkΨ−→ n→+∞0 , and let ε > 0 b e given. Let 0< δ ≤1 . Sine kgn/(εδ)kΨ−→ n→+∞0 , one has kgn/(εδ)kΨ≤1 , and hene RΩΨ|gn| εδ dP≤1 , for n large enough. Then, using the onvexity of Ψ : ZΩ Ψ|gn| εdP=ZΩ Ψδ|gn| εδ dP≤δZΩ Ψ|gn| εδ dP≤δ , for n large enough. Therefore, using Leb esgue's dominated onvergene Theorem, it follows that if Cφ:HΨ→HΨ is order b ounded into MΨ(T) , then the omp osition op erator Cφ:HΨ→HΨ is ompat. Pro of of Prop osition 3.14. As HMΨ is separable, there exists a sequene (fn)n≥1 in the unit ball of HMΨ suh that sup nfn◦φ(reiθ)=kδφ(reiθ)k(HMΨ)∗. Now, supp ose that Cφ is order b ounded into LΨ(T) (resp. into MΨ(T) ). Then there exists some g in LΨ(T) (resp. in MΨ(T) ) suh that g≥ |Cφ(f)| a.e. , for every f in the unit ball of HΨ . Using Lemma 3.11, we have a.e. Ψ−11 1−|φ(reiθ)|≤4kδφ(reiθ)k(HΨ)∗= 4 sup nfn◦φ(reiθ)≤4g. The result hene follows letting r↑1 . The onverse is obvious.  Theorem 3.15 If the omposition operator Cφ:HΨ→HΨ is order bounded, then: (OB3) m(1 −|φ|< λ) = O1 χA(1/λ), as λ→0, for some A > 0. and if it is order bounded into MΨ(T) , then: (OB4) m(1 −|φ|< λ) = O1 χA(1/λ), as λ→0, for every A > 0. Under the hypothesis Ψ∈∆1 , the onverse holds. 19 Pro of. The neessary ondition follows from Prop osition 3.14 and Markov's inequality: m1 1−|φ|> t≤1 χA(t)ZT χA1 1−|φ|dm. For the onverse, we shall prove a stronger result, and for that, we dene the weak- LΨ spae as follows: Denition 3.16 The weak- LΨ spae LΨ,∞(Ω) is the spae of measurable fun- tions f: Ω →C suh that, for some onstant c > 0 , one has, for every t > 0 : P(|f|> t)≤1 Ψ(ct)· For subsequent referenes, we shall state separately the following elementary result. Lemma 3.17 For every f∈LΨ(Ω) , one has, for every t > 0 : kfkΨ≥t Ψ−11 P(|f|>t)· Pro of. By Markov's inequality, one has, for t > 0 : Ψt kfkΨP(|f|> t)≤ZΩ Ψ|f| kfkΨdP≤1; and that gives the lemma.  Sine Lemma 3.17 an be read: P(|f|> t)≤1 Ψ(t/kfkΨ) , we get that LΨ(Ω) ⊆LΨ,∞(Ω) . The onverse of Theorem 3.15 is now an immediate onsequene of the following prop osition. Prop osition 3.18 1) a) If Ψ∈∆1 , then LΨ(Ω) = LΨ,∞(Ω) . b) If LΨ(Ω) = LΨ,∞(Ω) , then Ψ∈∆0 . 2) If LΨ(T) = LΨ,∞(T) , then ondition (OB3) implies that Cφ:HΨ→HΨ is order bounded, and ondition (OB4) implies that it is order bounded into MΨ(T) . Lemma 3.19 The fol lowing assertions are equivalent i) LΨ(Ω) = LΨ,∞(Ω) . ii) Z∞ 1 Ψ′(u) Ψ(Bu)du ≡Z+∞ Ψ(1) 1 χB(x)dx < +∞ , for some B > 1 . 20 Pro of of the Lemma. Assume rst that 1/χB is integrable on (Ψ(1),∞) . For every f∈LΨ,∞(Ω) , there is a c > 0 suh that P(|f|> t)≤1/Ψ(ct) . Then ZΩ Ψc|f| BdP=Z+∞ 0 P(|f|> Bt/c) Ψ′(t)dt ≤Ψ(1) + Z+∞ 1 P(|f|> Bt/c) Ψ′(t)dt ≤Ψ(1) + Z+∞ 1 Ψ′(t) Ψ(Bt)dt = Ψ(1) + Z+∞ Ψ(1) du χB(u)<+∞, so that f∈LΨ(Ω) . Conversely, assume that LΨ(Ω) = LΨ,∞(Ω) . Sine 1/Ψ is dereasing, there is a measurable funtion f: Ω →C suh that P(|f|> t) = 1/Ψ(t) , when t≥Ψ−1(1) . Suh a funtion is in LΨ,∞(Ω) ; hene it is in LΨ(Ω) , by our hyp othesis. Therefore, there is a B > 1 suh that ZΩ Ψ(|f|/B)dP<+∞. But ZΩ Ψ(|f|/B)dP=Z+∞ 0 P(|f|> Bt) Ψ′(t)dt ≥Z+∞ 1 BΨ−1(1) Ψ′(t) Ψ(Bt)dt ≥Z+∞ Ψ−1(1) Ψ′(t) Ψ(Bt)dt =Z+∞ 1 du χB(u) , and hene 1/χB is integrable on (1,∞) .  Pro of of Prop osition 3.18. 1)a) We rst remark that for every Orliz funtion Ψ , one has Ψ(2x)≥xΨ′(x) for every x > 0 , b eause, sine Ψ′ is p ositive and inreasing: Ψ(2x) = Z2x 0 Ψ′(t)dt ≥Z2x x Ψ′(t)dt ≥xΨ′(x). Assume now that Ψ∈∆1 : xΨ(x)≤Ψ(βx) for some β > 0 and for x≥x0>0 . By Lemma 3.19, it sues to show that 1/χ2β is integrable on (Ψ(x0),+∞) . But Z+∞ Ψ(x0) dx χ2β(x)=Z+∞ x0 Ψ′(u) Ψ(2βu)du ≤Z+∞ x0 Ψ(2u)/u 2uΨ(2u)du =Z+∞ x0 du 2u2<+∞. 1)b) Supp ose now that LΨ(Ω) = LΨ,∞(Ω) . By the preeding lemma, there exists some B > 1 suh that lim x→+∞Z2x x Ψ′(u) Ψ(Bu)du = 0. 21 By onvexity, Ψ(2x)≥2Ψ(x) and Ψ′ is nonnegative, so that Z2x x Ψ′(u) Ψ(Bu)du ≥1 Ψ(2Bx)Z2x x Ψ′(u)du ≥Ψ(2x)−Ψ(x) Ψ(2Bx)≥Ψ(x) Ψ(2Bx)· Therefore Ψ satises ∆0 . 2) Assume that LΨ(T) = LΨ,∞(T) and that ondition (OB3) (resp. (OB4)) holds. By Lemma 3.19, there is a B > 0 suh that 1/χB is integrable on (1,+∞) . We get, using ondition (OB3) (resp. (OB4)) and setting C=A/B : ZT χC1 1−|φ|dm =Z+∞ 0 m1 1−|φ|> tχ′ C(t)dt =Z+∞ 0 m(1 −|φ|<1/t)χ′ C(t)dt ≤χC(1) + KZ+∞ 1 χ′ C(t) χA(t)dt. But, if we set u=χC(t) , i.e. u= ΨCΨ−1(t) , then Ψ−1(u) = CΨ−1(t) , and hene χA(t) = ΨAΨ−1(t)= ΨBΨ−1(u)=χB(u). Therefore: ZT χC1 1−|φ|dm ≤χC(1) + KZ+∞ χC(1) du χB(u)du < +∞. It follows from Prop osition 3.14 that Cφ is order b ounded (resp. into MΨ(T) ).  Remark. The ondition Ψ∈∆1 is not equivalent to LΨ(Ω) = LΨ,∞(Ω) . For example, we an take: Ψ(x) = exp log(x+ 1)3/2−1. Then, as x tends to innity, Ψ(Kx)∼Ψ(x) exp 3 2(log K)(log x)1/2, and hene Ψ6∈ ∆1 . On the other hand, Ψ′(x)∼3 2 (log x)1/2 xΨ(x) ; hene: Z+∞ Ψ(1) du χK(u)=Z+∞ 1 Ψ′(t) Ψ(Kt)dt ∼Z+∞ 0 3u2 exp(3 2(log K)u)du =Z+∞ 0 3u2 K3u/2du < +∞ for K > 1 . Therefore LΨ(Ω) = LΨ,∞(Ω) by Lemma 3.19. 22 3.5 Weakly ompat omp osition op erators We saw in Lemma 3.10 that kua,rkΨ≤1 Ψ−1(1 1−r)· The next result shows that the weak ompatness of Cφ transforms the  big-oh  into a  little-oh , when Ψ grows fast enough. Theorem 3.20 Assume that Ψ∈∆0 . Then the weak ompatness of the omposition operator Cφ:HΨ→HΨ implies that: (W) sup a∈TkCφ(ua,r)kΨ=o1 Ψ−11 1−r, as r→1. Pro of. Atually, we only need to use that the restrition of Cφ:HΨ→HΨ to HMΨ is weakly ompat. We proved in [25℄, Theorem 4, that, under the hyp othesis Ψ∈∆0 , the op erator Cφ:HMΨ→HMΨ is weakly ompat if and only if for every ε > 0 , we an nd Kε>0 suh that, for every f∈HMΨ , one has: kCφ(f)kΨ≤Kεkfk1+εkfkΨ. Using Corollary 3.10, we get, for every ε > 0 , sine kua,rk1≤1−r : kCφ(ua,r)kΨ≤Kε(1 −r) + ε1 Ψ−1(1 1−r)· But Ψ(x) x−→ x→+∞+∞ . Hene 1−r=o1 Ψ−1(1 1−r) as r→1, and that proves the theorem.  We shall see in Setion 4 that the onverse holds when Ψ satises ∆0 and moreover that Cφ is then ompat. That will use some other tehniques. Nevertheless, we an prove, from now on, the following result. Theorem 3.21 If Ψ∈∆2 , then ondition: (W) sup a∈TkCφ(ua,r)kΨ=o1 Ψ−11 1−r, as r→1. in Theorem 3.20 implies ondition: (OB4) m(1 −|φ|< λ) = O1 χA(1/λ), as λ→0, for every A > 0. in Theorem 3.15 . 23 Remark that when Ψ∈∆0 (and in partiular when Ψ∈∆2 ), one has, for any B > βA (where β is given by the denition of ∆0 ): Ψ(Bx) Ψ(Ax)→+∞ as x→+∞ ; hene 1/χA(x) = o1/χB(x) and the  big-oh  ondition in (OB4) may b e replaed by a  little-oh  ondition. Before proving this theorem, let us note that: Prop osition 3.22 Condition (W) sup a∈TkCφ(ua,r)kΨ=o1 Ψ−11 1−r as r→1 implies that m(|φ|= 1) = 0 . Pro of. Otherwise, one has m(|φ|= 1) = δ > 0 . Splitting the unit irle T into N parts, we get some a∈T suh that: m(|φ−a| ≤ π/N)≥δ/N. But, the inequality |φ−a| ≤ π/N implies, with r= 1 −1/N (sine |φ| ≤ 1 ): |1−¯arφ| ≤ |1−¯aφ|+ (1 −r)|¯aφ|=|a−φ|+ (1 −r)|φ|<5/N = 5(1 −r). Hene: m|Cφ(ua,r)|>1/25)=m|1−¯arφ|<5(1 −r) ≥m|φ−a| ≤ π(1 −r)≥δ(1 −r), and therefore, by Lemma 3.17: kCφ(ua,r)kΨ≥1/25 Ψ−11/δ(1 −r)≥δ/25 Ψ−11/(1 −r)· Sine r an b e taken arbitrarily lose to 1 , that proves the proposition.  Pro of of Theorem 3.21. Assume that ondition (W) is satised, and x A > 1 . Let ε > 0 to b e adjusted later. We an nd rε<1 suh that rε≤r < 1 implies: kCφ(ua,r)kΨ≤ε Ψ−11 1−r , ∀a∈T. Now, Lemma 3.17 also reads: m(|f|> t)≤1 Ψt kfkΨ , so that, if one sets B= 1/9ε : m|1−¯arφ|<3(1 −r)=m|Cφ(ua,r)|>1/9≤1 ΨBΨ−11 1−r· We laim that this implies a go o d upp er b ound on m(|φ|> r) , even if we lo ose a fator 1/(1 −r) , due to the eet of a rotation on φ . For that, we shall use the following lemma. 24 Lemma 3.23 Let φ:D→D be an analyti funtion. Then, for every r with 0< r < 1 , there exists a∈T suh that: m|1−¯ar φ|<3(1 −r)≥1−r 8m(|φ|> r). Admitting this for a while, we are going to nish the pro of. Fix an r suh that rε≤r < 1 , and take an a∈T as in Lemma 3.23. We get, from the preeding, in setting λ= 1 −r : m(1 −|φ|< λ) = m(|φ|> r) ≤8 1−rm|1−¯arφ|<3(1 −r) ≤8 1−r 1 ΨBΨ−11 1−r , i.e. , setting x= Ψ−1(1/1−r) = Ψ−1(1/λ) : m(1 −|φ|< λ)≤8Ψ(x) Ψ(Bx)· But Ψ saties the ∆2 -ondition: Ψ(y)2≤Ψ(αy) for some α > 1 and y large enough. Then, adjusting now ε > 0 as ε= 1/9αA , in order that B=αA , we get, for x large enough, sine A > 1 : Ψ(x)Ψ(Ax)≤[Ψ(Ax)]2≤Ψ(Bx). Therefore, for r lose enough to 1 : m(1 −|φ|< λ)≤8 Ψ(Ax)=8 χA(1/λ)· We hene get ondition (OB4), and that proves Theorem 3.21.  Pro of of Lemma 3.23. Let λ= 1 −r , and let δ > 0 b e a numb er whih we shall sp eify later. Consider the set: Cδ={z∈D;|z| ≥ 1−λ and |arg z| ≤ δ} (for δ=λ , Cδ is a losed Carleson window). It is geometrially lear that (1 −λ)Cδ is ontained in the losed disk of enter 1 and whose edge ontains (1 −λ)2eiδ ; hene, for every z∈Cδ , one has: |1−(1 −λ)z|2≤ |1−(1 −λ)2eiδ|2= 2(1 −λ)2(1 −cos δ) + λ2(2 −λ)2 ≤(1 −λ)2δ2+λ2(2 −λ)2≤9λ2 if δ≤λ . 25 Therefore: ZT Ψ|˘g| 2αdm = +∞ X n=1 Z˜ An Ψ|˘g| 2αdm. Sine ˘gj= 0 on ˜ An for j > n , we get: ZT Ψ|˘g| 2αdm = +∞ X n=1 Z˜ An Ψ|˘g1|+···+|˘gn| 2αdm. Now, by the onvexity of Ψ : Ψ|˘g1|+···+|˘gn| 2α≤1 2Ψ|˘g1|+···+|˘gn−1| α+ Ψ|˘gn| α· But: Ψ|˘gn| α≤1 2nΨ(|˘gn|) + Ψ(xn), and Ψ|˘g1|+···+|˘gn−1| α≤Ψt1+···+tn−1 α; therefore, using that RTΨ(|˘gn|)dm ≤RTΨ(|gn|)dm ≤1 : ZT Ψ|˘g| 2αdm ≤ +∞ X n=1 1 2Ψt1+···+tn−1 αm(˜ An) +1 2nZT Ψ(|˘gn|)dm + Ψ(xn)m(˜ An) ≤ +∞ X n=1 1 2hΨt1+···+tn−1 α+ Ψ(xn)i2ntn Ψ(tn)+1 2n ≤ +∞ X n=1 1 2h1 2n+1 2ni= 1 , whih proves that ˘g∈LΨ , and k˘gkΨ≤2α . The pro of is fully ahieved.  Remark. In the ab ove pro of, we hose Mn= log(n+ 1) . This hoie was only used to onlude that PnkCφ(gn)kp Ψ= +∞ for every p < ∞ . Therefore, the ab ove pro of shows that, given any inreasing funtion Υ: (0,∞)→(0,∞) tending to ∞ , we an nd, with a suitable hoie of a slowly inreasing sequene (Mn)n≥1 , a symb ol φ and a sequene (gn)n≥1 in HΨ suh that Pn|gn| ∈ LΨ , although PnΥ(kCφ(gn)kΨ) = +∞ . 32 4 Carleson measures 4.1 Intro dution B. MaCluer ([27℄; see also [11℄, Theorem 3.12) has haraterized ompat omp osition op erators on Hardy spaes Hp ( p < ∞ ) in term of Carleson measures. In this setion, we shall give an analogue of this result for Hardy-Orliz spaes HΨ , but in terms of  Ψ -Carleson measures. Indeed, Carleson measures do not haraterize the ompatness of omp osition op erators when Ψ grows to o quikly, as it follows from Corollary 3.26. Before that, we shall reall some denitions (see for example [11℄, pages 3738, or [12℄, page 157). Let ξ∈T and h∈(0,1) . Dene (4.1) S(ξ, h) = {z∈D;|ξ−z|< h}. The Carleson window W(ξ, h) is the following subset of D : (4.2) W(ξ, h) = {z∈D; 1 −h < |z| ≤ 1 and |arg(z¯ ξ)|< h}. It is easy to show that we have for every ξ∈T and h∈(0,1) : S(ξ, h/2) ⊆W(ξ, h) and W(ξ, h/2) ⊆S(ξ, h), so that, in the sequel, we may work equivalently with either S(ξ, h) or W(ξ, h) . Reall that a p ositive Borel measure µ on D (or D ) is alled a Carleson measure if there exists some onstant K > 0 suh that: µS(ξ, h)≤Kh , ∀ξ∈T,∀h∈(0,1). Carleson's Theorem (see [11℄, Theorem 2.33, or [12℄, Theorem 9.3) asserts that, for 0< p < ∞ , the Hardy spae Hp is ontinuously embedded into Lp(µ) if and only if µ is a Carleson measure. Given an analyti self-map φ:D→D , we dene the pul lbak measure µφ on the losed unit disk D (whih we shall denote simply µ when this is unambiguous) as the image of the Haar measure m of T=∂D under the map φ∗ (the b oundary limit of φ ): (4.3) µφ(E) = mφ∗−1(E), for every Borel subset E of D . The automati ontinuity of omp osition op erators Cφ on the Hardy spae Hp , ombined with Carleson's Theorem means that µφ is always a Carleson measure. B. MaCluer ([27℄, [11℄, Theorem 3.12) showed that: The omposition operator Cφ is ompat on H2 if and only if: µφS(ξ, h)=o(h) as h→0, uniformly for ξ∈T. (MC) 33 While the Shapiro's ompatness riterion, via the Nevanlinna ounting fun- tion ([36℄), deals with the b ehavior of φ inside the op en unit disk, the hara- terization (MC) deals with its b oundary values φ∗ . It is natural to wonder whether the mo dulus of φ∗ on T=∂D sues to haraterize the ompatness of Cφ . This leads to the following question: if two funtions φ1 and φ2 have the same mo dulus on T , are the ompatness of the two asso iated omp osition op erators equivalent? We have seen in Theorem 3.24 that the answer is p ositive on HΨ when Ψ∈∆2 . However, on H2 it turns out to be negative. We give the following ounterexample. Theorem 4.1 There exist two analyti funtions φ1 and φ2 from D into itself suh that |φ∗ 1|=|φ∗ 2| on T but for whih the omposition operator Cφ2:H2→H2 is ompat, though Cφ1:H2→H2 is not ompat. Remark. Let Ψ b e an Orliz funtion whih satises ∆2 . We shall see in Theorem 4.3 that every omposition op erator Cφ:H2→H2 is ompat as so on as Cφ:HΨ→HΨ is ompat. Hene, in the ab ove theorem, Cφ1:HΨ→HΨ is not ompat. It follows hene from Theorem 3.24, sine φ∗ 1 and φ∗ 2 have the same mo dulus, that Cφ2:HΨ→HΨ is not ompat (and, even, not weakly ompat), though Cφ2:H2→H2 is ompat. We have already seen suh a phenomenon in Corollary 3.26. However, the results of the next subsetion will allow us to onlude (Theorem 4.22) that, when Ψ(x) = exp log(x+ 1)2−1 , whih do es not satisfy ondition ∆2 , but satises onditions ∆1 and ∇1 , the omp osition op erator Cφ2:HΨ→HΨ is ompat, but not order b ounded into MΨ(T) . That will show that our assumption that Ψ∈∆2 in Theorem 3.24 is not only a tehnial one. Pro of. We start simply with φ1(z) = 1+z 2 . It is well known that Cφ1 is not ompat on H2 (this was rst observed in H. J. Shwartz's thesis [34℄: see [39℄, page 471). Now, let: M(z) = exp −1 + z 1−z and φ2(z) = φ1(z)M(z). For simpliity, we shall write φ=φ2 , and we are going to show that Cφ is a ompat op erator on H2 , using the riterion (MC) . Let ξ= eiα ∈T , with |α| ≤ π . We are going to prove that: µφS(ξ, h)=O(h3/2). First, observe that, for h∈(0,1) : S(ξ, h)⊆ {z∈D; 1 −h < |z| ≤ 1 and |arg(¯zξ)| ≤ 2h}. Hene for h small enough, µφS(ξ,h)≤ m{θ∈(−π, π) ; 1 −h < |φ(eiθ)| ≤ 1 and |arg(e−iαφ(eiθ))| ≤ 2h}). 34 For θ∈(−π, π) , one has |φ(eiθ)|=|φ1(eiθ)|= cos(θ/2) , and so the ondition 1−h < |φ(eiθ)| ≤ 1 is equivalent to 1−h < cos(θ/2) <1 , whih implies, sine cos t= 1 −2 sin2(t/2) ≤1−2t2/π2≤1−t2/5 for 0≤t≤π/2 (b eause sin t≥2 πt ), that 1−h < 1−(θ/2)2/5 , i.e. θ2≤20h and so |θ| ≤ 6√h . On the other hand, M(eiθ) = exp −icot(θ/2) ; hene arg φ(eiθ) = θ/2−cot(θ/2) , mo dulo 2π . Therefore, for h small enough: µφS(ξ, h)≤m({|θ| ≤ 6√h;|−α+θ/2−cot(θ/2)| ≤ 2h, mod 2π}) ≤2X n∈Z m({|t| ≤ 3√h;|−α+t−cot t+ 2πn| ≤ 2h}). We have to ma jorize b oth X n∈Z m({0< t ≤3√h;|−α+t−cot t+ 2πn| ≤ 2h}) and X n∈Z m({0< t ≤3√h;|α+t−cot t+ 2πn| ≤ 2h}) . The funtion F(t) = t−cot(t) is inreasing , and we dene an, bn∈(0, π) by: F(an) = α−2π(nh+n)−2h and F(bn) = α−2π(nh+n) + 2h, where the integer nh is given large enough to ensure that a0≤3√h . Of ourse, an< bn< an−1 . Observe that 4h=F(b0)−F(a0)≥b0−a0 , and then b0≤3√h+ 4h≤4√h for h small enough. One has: X n∈Z m({0< t ≤3√h;|−α+t−cot t+ 2πn| ≤ 2h})≤∞ X n=0 (bn−an). Sine F′(t) = 1 + 1 sin2t≥1 t2 , one has, on the one hand: (4.4) 4h=F(bn)−F(an) = Zbn an F′(t)dt ≥bn−an anbn ; hene: (4.5) bn−an≤4hanbn≤4hb2 n, for all n≥0. On the other hand, let us rst p oint out that, for 0≤t≤1 , F′(t) = 1 + 1 sin2t≤1 + π2/4 t2≤4 t2· Hene, for h small enough: 2π=F(bn)−F(bn+1) = Zbn bn+1 F′(t)dt ≤4bn−bn+1 bn+1bn , 35 and we get: b2 n+1 ≤bn+1bn≤2 π(bn−bn+1). Hene, using the fat that (4.4) gives b0−a0≤4ha0b0≤48 h2 , we get, from (4.5): ∞ X n=0 (bn−an)≤(b0−a0) + 8h π ∞ X n=0 (bn−bn+1) ≤48 h2+8h πb0≤48 h2+8h π4√h≤48√h+32 πh3/2≤11 h3/2, for h small enough. In the same way, we have ∞ X n=0 m({0< t ≤3√h;|α+t−cot t+ 2πn| ≤ 2h})≤11 h3/2. We an hene onlude that µφS(ξ, h)≤Ch3/2 , where C is a numerial onstant.  Remark. Cφ atually maps ontinuously H2 into H3 , and ompatly H2 into Hp , for any p < 3 (see [16℄ or Theorem 4.10 and 4.11). However, in some ases, the b ehaviour of |φ∗| on the b oundary ∂D sues. Prop osition 4.2 Let φ1 and φ2 be two analyti self-maps of D suh that |φ∗ 1| ≤ |φ∗ 2| on ∂D . Assume that they are both one-to-one on D , and that there exists a∈D suh that φ2(a) = 0 . Then the ompatness of Cφ2:H2→H2 implies that of Cφ1:H2→H2 . Pro of. By omp osing φ1 and φ2 with the automorphism of D whih maps 0 into a , we may assume that a= 0 . We an hene write φ2(z) = zφ(z) , with φ:D→C analyti in D . φ do es not vanish in D b eause of the injetivity of φ2 (this is obvious for z6= 0 , and for z= 0 , follows from the fat that the injetivity of φ2 implies φ′ 2(0) 6= 0 ). Then there is some δ > 0 suh that |φ(z)| ≥ δ for every z∈D . In fat, by ontinuity, there is some α > 0 and some 0< r < 1 suh that |φ(z)| ≥ α and |φ2(z)| ≤ α for |z| ≤ r . But b eing analyti and non onstant, φ2 is an op en map, so there is some ρ > 0 suh that ρD⊆φ2(rD) . Injetivity of φ2 shows that φ2(D\rD)∩ρD=∅ , that is to say that |φ2(z)| ≥ ρ for |z|> r . A fortiori |φ(z)| ≥ ρ for |z|> r . The laim is proved with δ= min(α, ρ) . Then 1 φ∈H∞ , as well as φ1 φ . Sine φ∗ 1 φ∗=φ∗ 1 φ∗ 2≤1 , one has φ1(z) φ(z)≤1 for every z∈D . Hene: 1−|φ1(z)| 1−|z|≥1−|φ(z)| 1−|z|=1−|φ2(z)| 1−|z|−|φ(z)|. 36 Now ([36℄, Theorem 3.5), the ompatness of Cφ2:H2→H2 implies that lim z< →1 1−|φ2(z)| 1−|z|= +∞; we get: lim z< →1 1−|φ1(z)| 1−|z|= +∞, whih implies the ompatness of Cφ1:H2→H2 , thanks to the injetivity of φ1 ([36℄, Theorem 3.2).  4.2 Compatness on HΨ versus ompatness on H2 The equivalene (MC) holds atually for every Hp spae (with p < ∞ ) instead of H2 . We are going to see in this setion that for Hardy-Orliz spaes HΨ , one needs a new notion of Carleson measures, whih one may all Ψ - Carleson measures . Before that, we are going to see that ondition (MC) allows to get that the ompatness of omp osition op erators on HΨ always implies that on Hp for p < ∞ . Reall that, when Ψ∈∆2 , we have seen in Corollary 3.26 that the onverse is not true. Theorem 4.3 Let φ:D→D be an analyti funtion. If one of the fol lowing onditions: i) Cφ is a ompat operator on HΨ ii) Ψ∈∆0 and Cφ is a weakly ompat operator on HΨ is satised, the omposition operator Cφ is ompat on H2 . Note that we have proved in Theorem 3.24 that the weak ompatness of Cφ:HΨ→HΨ implies its ompatness only when Ψ satises the ∆2 ondition. Nevertheless, we shall show in Theorem 4.21 that when Ψ∈∆0 , the weak ompatness of Cφ is equivalent to its ompatness. This is obviously false (in partiular when LΨ is reexive) without any assumption on Ψ . Pro of. We are going to use the haraterization (MC) for ompat omp osition op erators on H2 . Supp ose that the ondition on µφ is not fullled. Then there exist β∈(0,1) , ξn∈T , and hn∈(0,1) , with hn−→ n→+∞0 , suh that: µφS(ξn, hn)≥βhn. We are now going to use the funtion: vn(z) = h2 n (1 −anz)2 , 37 where: an= (1 −hn)ξn. Of ourse, vn is atually nothing but uξn,1−hn . We have by Corollary 3.10: kvnkΨ≤1 Ψ−1(1/hn)· Dene gn= Ψ−1(1/hn)vn , whih is in the unit ball of HMΨ . We have assumed at the b eginning of the pap er that x=oΨ(x) as x→ ∞ ; hene Ψ−1(x) = o(x) as x→ ∞ , and so h2 nΨ−1(1/hn)→0 . Therefore (gn)n onverges uniformly to zero on ompat subsets of D and kgnk1→0 , b eause kgnk1≤hnΨ−1(1/hn) . Then, in b oth ases, we should have kCφ(gn)kΨ→0 . Indeed, in ase i) , this follows from Prop osition 3.6, and in ase ii) , this follows from [25℄, Theorem 4. We are going to show that this is not true. Indeed: ZT Ψ4 β|gn◦φ|dm ≥ZD Ψ4 βΨ−1(1/hn)|vn(z)|dµφ ≥ZS(ξn,hn) Ψ4 βΨ−1(1/hn)|vn(z)|dµφ. But when z∈S(ξn, hn) , one has |vn(z)| ≥ 1/4 , b eause |1−anz| ≤ |1−anξn|+|an(ξn−z)|=hn+ (1 −hn)hn≤2hn. We obtain that by onvexity (sine β < 1 ): ZT Ψ4 β|gn◦φ|dm ≥ZT 1 βΨ4Ψ−1(1/hn)|vn(z)|)dm ≥1 βhn µφS(ξn, hn)≥1. This implies that kCφ(gn)kΨ≥β/4 and proves the theorem.  4.3 General measures We used several times the riterion (MC) for ompatness on H2 via Carleson measures. The fat that this provides suh a useful tool leads to wonder if the b oundedness and the ompatness on Hardy-Orliz spaes an b e expressed in suh a pleasant manner. Theorem 4.10 and Theorem 4.11 are the Orliz version of Carleson's Theorem for Hp spaes ([11℄, Theorem 2.35). The key for our general haraterization is the use of the following funtions (see (4.2) for the denition of the Carleson's window W(ξ, h) ). 38 Denition 4.4 For any positive nite Borel measure µ on the unit disk D (or on D ), we set, for h∈(0,1] : ρµ(h) = sup ξ∈T µW(ξ, h), (4.6) Kµ(h) = sup 0<t≤h ρµ(t) t (4.7) Hene µW(ξ, t)≤Kµ(h)t for t≤h . The measure µ is a Carleson measure if and only if Kµ(h) is b ounded by a onstant K , for h∈(0,1) and this happ ens as so on as Kµ(h0) is nite for some h0∈(0,1) . Denition 4.5 We say that Ψ satises the ∇0 ondition if for some x0>0 , C≥1 and every x0≤x≤y , one has: (4.8) Ψ(2x) Ψ(x)≤Ψ(2Cy) Ψ(y)· This is a ondition on the regularity of Ψ . It is satised if Ψ(2x) Ψ(x)≤CΨ(2y) Ψ(y)· Prop osition 4.6 the fol lowing assertions are equivalent i) Ψ satises the ∇0 ondition. ii) There exists some x0>0 satisfying: for every β > 1 , there exists Cβ≥1 suh that Ψ(βx) Ψ(x)≤Ψ(βCβy) Ψ(y) , for every x0≤x≤y. iii) There exist x0>0 , β > 1 and Cβ≥1 suh that Ψ(βx) Ψ(x)≤Ψ(βCβy) Ψ(y) , for every x0≤x≤y. Pro of. We only have to prove i)⇒ii) , sine iii)⇒i) is similar and ii)⇒iii) is trivial. If β∈(1,2] , it is easy, taking Cβ= 2C/β . Now, if β∈(2b,2b+1] for some integer b≥1 , we write for every x0≤x≤y : Ψ(βx) Ψ(x)≤Ψ(2b+1x) Ψ(x)=Ψ(2b+1x) Ψ(2bx)···Ψ(2x) Ψ(x)· 39 But we have for every integer j≥1 : 2j−1x≤(2C)j−1x≤(2C)j−1y , so: Ψ(2jx) Ψ(2j−1x)≤Ψ((2C)jy) Ψ((2C)j−1y) , and we obtain: Ψ(βx) Ψ(x)≤Ψ((2C)b+1y) Ψ(y)≤Ψ(βCβy) Ψ(y) , where Cβ= 2Cb+1 .  Examples. It is immediately seen that the following funtions satisfy ∇0 : Ψ(x) = xp , Ψ(x) = exp log(x+ 1)α−1 , Ψ(x) = exα−1 , α≥1 . Note that when Ψ∈ ∇0 , with onstant C= 1 , i.e. Ψ(βx)/Ψ(x) is inreasing for x large enough, then we have the dihotomy: either Ψ∈∆2 , or Ψ∈∆0 . We shall say that ∇0 is uniformly satised if there exist C≥1 and x0>0 suh that, for every β > 1 : (4.9) Ψ(βx) Ψ(x)≤Ψ(Cβy) Ψ(y) for x0≤x≤y, One has: Prop osition 4.7 1) Condition ∆2 implies ondition ∇0 uniformly. 2) If Ψ∈ ∇0 uniformly, then Ψ∈ ∇1 . 3) The funtion κ(x) = log Ψ(ex) is onvex on (x0,+∞) if and only if ∇0 is satised with onstant C= 1 . We shall say that Ψ is κ -onvex when κ is onvex at innity. Note that Ψ is κ -onvex whenever Ψ is log-onvex. In the ab ove examples Ψ is κ -onvex; it also the ase of Ψ(x) = x2/log x , x≥e ; but, on the other hand, if Ψ(x) = x2log x for x≥e , then Ψ is not κ -onvex. Nevertheless, for β2≤x≤y , one has: Ψ(βx) Ψ(x)=β21 + log β log x≤3β2 2≤3 2 Ψ(βy) Ψ(y) , and hene Ψ∈ ∇0 . We do not know whether Ψ∈ ∇0 uniformly implies that Ψ is equivalent to an Orliz funtion for whih the asso iated funtion κ is onvex. Pro of. 1) Sine Ψ∈∆2 , one has Ψ(u)2≤Ψ(αu) for some α > 1 and x≥x0 . We may assume that Ψ(x0)≥1 . Then, for y≥x≥x0 and every β > 1 : Ψ(βx)Ψ(y)≤Ψ(βy)2≤Ψ(αβy)≤Ψ(αβy)Ψ(x), whih is (4.9). 40 2) Supp ose that Ψ satises ondition ∇0 uniformly. We may assume that Ψ(x0)≥1 . Let x0≤u≤v ; we an write u=βx0 for some β≥1 . Then ondition (4.9) gives: Ψ(u)Ψ(v) = Ψ(βx0)Ψ(v)≤Ψ(x0)Ψ(Cβv)≤ΨΨ(x0)Cβv= Ψ(buv), with b=CΨ(x0)/x0 . 3) Assume that κ is onvex on (x0,+∞) . For every β > 1 , let κβ(t) = κ(tlog β) = log(Ψ(βt)) , whih is onvex on (x0/log(β),+∞) . Taking y≥x≥ ex0 , write x=βθ and y=βθ′ with θ≤θ′ . Convexity of κ gives, sine θ′≥θ≥ x0/log(β) : κβ(θ+ 1) −κβ(θ)≤κβ(θ′+ 1) −κβ(θ′), whih means that: Ψ(βx) Ψ(x)≤Ψ(βy) Ψ(y)· Assume that (4.9) is fullled for every β > 1 , with C= 1 . Then, taking y=βx , one has: Ψ(βx)2≤Ψ(x)Ψ(β2x). Let u < v b e large enough. Taking x=u2 and β=v/u , we get: Ψ(uv)2≤Ψ(u2)Ψ(v2), whih means that κ is onvex.  Remark. The growth and regularity onditions for Ψ an b e expressed in the following form: •Ψ∈∆0 i κ(x+β′)−κ(x)−→ x→∞+∞ , for some β′>0 . •Ψ∈∆1 i for some β′>0 , one has x+κ(x)≤κ(x+β′) , for x large enough. •Ψ∈∆2 i for some α′>0 , one has 2κ(x)≤κ(x+α′) , for x large enough. •Ψ∈ ∇1 i κB(x)+κB(y)≤κB(x+y) for x, y large enough, with B= e−b . •Ψ∈ ∇0 i for some c′≥1 and A > 1 , one has κA(θ+ 1) −κA(θ)≤ κA(θ′+c′)−κA(θ′) for θ≤θ′ large enough. Before proving the main results of this setion, let us ollet some basi fats on the ompatness of the embedding of HΨ1 into LΨ2(µ) . First: 41 Z∞ xA Ψ′ 2(t) Ψ2(2t)Ψ1(2t/A)m({Mf>8t/A})dt ≤Z∞ 0 Ψ1(2t/A) tm({Mf>8t/A})dt ≤2 AZ∞ 0 Ψ′ 1(2t/A)m({Mf>8t/A})dt =Z∞ 0 Ψ′ 1(x)m({Mf>4x})dx =ZT Ψ11 4Mfdm ≤1 4ZT Ψ1(Mf)dm. whih leads to the desired result.  For the pro ofs of Theorem 4.10 and Theorem 4.11, we may restrit ourselves to the ase of funtions Ψ1 and Ψ2 satisfying ∇2 . Indeed, supp ose that Ψ1 and Ψ2 are Orliz funtions and dene f Ψj(t) = Ψj(t2) , for j∈ {1,2} . The funtions f Ψ1 and f Ψ2 are Orliz funtions satisfying ∇2 sine, with β= 2 , we have for every t≥0 : f Ψj(βt) = Ψj(4t2)≥4Ψj(t2) = 2βf Ψj(t). Now, we laim that µ satises ( R ), ( K ), ( R0 ) or ( K0 ) for the ouple Ψ1,Ψ2 if and only if µ satises it for the ouple f Ψ1,f Ψ2 . This is simply due to the fat that for every A > 0 and t≥0 , we have f Ψ2Af Ψ1−1(t)= Ψ2A2Ψ−1 1(t). Moreover, notie that, writing f=Bg2 (where B is a Blashke pro dut), we have f∈HΨ if and only if g∈He Ψ ; thus kgkLe Ψ=pkfkLΨ . It is then lear that Id :HΨ1−→ LΨ2(µ)=Id :Hf Ψ1−→ Lf Ψ2(µ) 2, so that the anonial emb edding is b ounded (resp. ompat) for the ouple Ψ1,Ψ2 if and only if it is so for the ouple f Ψ1,f Ψ2 , thanks to Prop osition 4.9. Pro of of Theorem 4.10. 1) Let C b e the norm of the anonial emb edding j:HΨ1֒→LΨ2(µ) , and let ξ∈T and h∈(0,1) . It sues to test the ontinuity of j on f= Ψ−1 1(1/h)uξ,1−h , whih is in the unit ball of HMΨ1 , by Corollary 3.10. But, when z∈W(ξ, h) one has, with a= (1 −h)ξ : |1−¯az| ≤ |1−¯aξ|+|¯aξ −¯az|=h+ (1 −h)hξ−z |z|+z |z|−zi ≤h+ (1 −h)[h+ (1 −|z|)] ≤h+ (1 −h)[h+h]≤3h; 48 hene |uξ,1−h(z)| ≥ 1/9 and |f(z)| ≥ (1/9)Ψ−1 1(1/h) ; therefore: 1≥ZD Ψ2|f| Cdµ ≥Ψ21 9CΨ−1 1(1/h)µW(ξ, h), whih is ( R ). 2) By Prop osition 3.5, the maximal (non-tangential) funtion M is b ounded on LΨ1(T) : there exists a onstant C≥1 suh that kMfkΨ1≤CkfkΨ1 for every f∈LΨ1(T) . We x f in the unit ball of HΨ1 (note that kf/CkΨ1 remains ≤1 ) and use Lemma 4.14, with E=D and f replaed by f/C (here hA= 1 )). Writing ˜ C=π 2+µ(D)Ψ2(xA) , we get: ZD Ψ2A 8C˜ C|f|dµ ≤1 ˜ CZD Ψ2A 8C|f|dµ ≤1 ˜ C µ(D)Ψ2(xA) + π 2ZT Ψ11 CMfdm! ≤1 ˜ Cµ(D)Ψ2(xA) + π 2= 1, whih means that kfkLΨ2(µ)≤8C˜ C A· Pro of of Theorem 4.11. 1) Supp ose that the emb edding is ompat, but that ondition ( R0 ) is not satised. Then there exist ε0∈(0,1) , A > 0 , a sequene of p ositive numbers (hn)n dereasing to 0 , and a sequene of ξn∈T , suh that: µW(ξn, hn)≥ε0 Ψ2AΨ−1 1(1/hn)· Consider the sequene of funtions fn(z) = Ψ1−1(1/hn)h2 n (1 −¯anz)2= Ψ1−1(1/hn)uξn,|an|, where an= (1 −hn)ξn . By Corollary 3.10, fn is in the unit ball of HMΨ1 . Moreover, it is plain that (fn)n onverges to 0 uniformly on every ompat subset of D . By the ompatness riterion (Prop osition 3.6), (fn)n is norm- onverging to 0 in LΨ2(µ) . But, as ab ove (pro of of Theorem 4.10), for every n≥1 , one has |fn(z)| ≥ (1/9)Ψ−1 1(1/hn) when z∈W(ξn, hn) ; hene: ZD Ψ29A ε0|fn|dµ ≥Ψ2A ε0 Ψ−1 1(1/hn)µW(ξn, hn) ≥Ψ2A ε0 Ψ−1 1(1/hn)ε0 Ψ2AΨ−1 1(1/hn)≥1, 49 by the onvexity of Ψ2 . This implies that kfnkLΨ2(µ)≥ε0/9A and gives a ontradition. 2) We have to prove that for every ε > 0 , there exists an r∈(0,1) suh that the norm of the injetion Ir:HΨ1−→ LΨ2(D\rD, µ) is smaller than ε (see Prop osition 4.9). Let C≥1 b e the norm of the maximal op erator, as in the pro of of Theorem 4.10: kMfkΨ1≤CkfkΨ1 for every f∈LΨ1(T) , and set A= 16C/ε . Condition ( K0 ) gives us hA∈(0,1) suh that: Kµ(h)≤1 2 1/h Ψ2AΨ−1 1(1/h) when h≤hA . Let f in the unit ball of HΨ1 and r∈(0,1) . By Lemma 4.14: ZD\rD Ψ2|f| εdµ =ZD\rD Ψ2A 16C|f|dµ ≤1 2ZD\rD Ψ2A 8C|f|dµ ≤1 2 µD\rDΨ2(xA) + π 2ZT Ψ1Mf Cdm! ≤π 4+ Ψ2(xA)µD\rD. As µ(T) = 0 , there exists some r0∈(0,1) suh that π 4+ Ψ2(xA)µD\rD≤1 , for every r∈(r0,1) . This ends the pro of of 2). 3) Assume that Ψ satises ondition ∇0 and that ondition ( R0 ) is fullled: for every A > 0 and every h∈(0, hA) ( hA small enough), we have: ρµ(h)≤1 ΨAΨ−1(1/h)· This implies that: Kµ(h) = sup 0<s≤h ρµ(s) s≤sup 0<s≤h 1/s ΨAΨ−1(1/s)= sup x≥Ψ−1(1/h) Ψ(x) Ψ(A x)· Fix an arbitrary β > 1 and ho ose A=βCβ>1 , where Cβ is given by the ∇0 ondition for Ψ and Prop osition 4.6. We have, for h small enough and x≥Ψ−1(1/h) : ΨβΨ−1(1/h) ΨΨ−1(1/h)≤Ψ(βCβx) Ψ(x)=Ψ(A x) Ψ(x); we get hene, for h small enough: Kµ(h)≤ΨΨ−1(1/h) ΨβΨ−1(1/h)=1/h ΨβΨ−1(1/h) , 50 and ondition ( K0 ) is fullled. With 1) and 2) previously shown, this nishes the pro of.  Remark. Atually, the pro of of Theorem 4.10, 1) gives, for every measure µ on D and every Orliz funtion Ψ : 1≥ZD Ψ|uξ,1−h| kuξ,1−hkLΨ(µ)dµ ≥Ψ1 9kuξ,1−hkLΨ(µ)µW(ξ, h), and hene: (4.12) µW(ξ, h)≤1 Ψ1 9kuξ,1−hkLΨ(µ)· In partiular, if µ=µφ is the image of the Haar measure m under a self-map φ of D , one has: (4.13) µW(ξ, h)≤1 Ψ1 9kCφ(uξ,1−h)kΨ· Condition (4.12) allows to have an upp er ontrol for the µ -measure of Carleson windows, with kuξ,1−hkLΨ(µ) . It is p ossible, onversely, to ma jorize these norms. Denition 4.15 We shal l say that a measure µ on D is a Ψ -Carleson measure if there exists some A > 0 suh that µW(ξ, h)≤1 ΨAΨ−1(1/h) , for every ξ∈T and every h∈(0,1). We shal l say that a measure µ on D is a vanishing Ψ -Carleson measure if, for every A > 0 , lim h→0ΨAΨ−1(1/h).ρµ(h) = 0. Equivalently if, for every A > 0 , there exists hA∈(0,1) suh that µW(ξ, h)≤1 ΨAΨ−1(1/h) , for every ξ∈T and every h∈(0, hA). We have the following haraterizations: Prop osition 4.16 1) µ is a Ψ -Carleson measure on D if and only if there exists some onstant C≥1 suh that : kuξ,1−hkLΨ(µ)≤C Ψ−1(1/h) , for every ξ∈T and every h∈(0,1). 2) µ is a vanishing Ψ -Carleson measure on D if and only if lim h→0sup ξ∈T Ψ−1(1/h)kuξ,1−hkLΨ(µ)= 0. 51 Pro of. The suieny (b oth for 1) and 2)) follows easily from (4.12) in the preeding remark. The onverse is an obvious onsequene of the following lemma. Lemma 4.17 Suppose that there exist A > 0 and h0∈(0,1) suh that: ρµ(h)≤1 ΨAΨ−1(1/h) , for every h∈(0, h0). Then there exists h1∈(0,1) suh that: kuξ,1−hkLΨ(µ)≤24 AΨ−1(1/h) , for every ξ∈T and every h∈(0, h1) . Pro of of the lemma. This is inspired from [13℄, Chapter VI, Lemma 3.3, page 239. We may assume that h≤h0/4≤1/4 . First, writing a= (1−h)ξ (where ξ∈T ), we observe that, when |z−ξ| ≥ bh for a b > 0 , we have: |1−¯az|2= 1 + |a|2|z|2−2|a|ℜ(¯ ξz) =|a||ξ−z|2+ (1 −|a|) + |a|2|z|2−|a||z|2 ≥ |a|b2h2+ (1 −|a|)2≥(|a|b2+ 1)h2. So, we have |uξ,1−h(z)| ≤ 1 |a|b2+ 1 ≤min(1,2/b2) , when |z−ξ| ≥ bh . Now, dene, for every n∈N and ξ∈T : Sn=S(ξ, 2n+1h) = {z∈D;|z−ξ|<2n+1h} ⊂ W(ξ, 2.2n+1h). Our observation implies that |uξ,1−h(z)| ≤ min(1,2/4n) , for every z∈D\Sn−1 . For z∈S0 , one has simply |uξ,1−h(z)| ≤ 1 . There exists an integer N suh that 2N+2h≤h0<2N+3h . Let us ompute: ZD ΨM|uξ,1−h|dµ =ZS0 ΨM|uξ,1−h|dµ + N X n=1 ZSn\Sn−1 ΨM|uξ,1−h|dµ +ZD\SN ΨM|uξ,1−h|dµ ≤Ψ(M)µ(S0) + N X n=1 Ψ2M 4nµ(Sn) + Ψ2M 4Nµ(D) ≤ N X n=0 1 2n+1 Ψ4M 2nµ(Sn) + Ψ2M 4Nµ(D). 52 But for n≤N , we have 2.2n+1h≤2N+2h≤h0 , so the hypothesis gives: µ(Sn)≤1 ΨAΨ−1(1/2n+2h)· Take now: (4.14) M=A 24Ψ−11 h· We have, using that Ψ−11 2n+2h≥1 2n+2 Ψ−11 h , ZD ΨM|uξ,1−h|dµ ≤ N X n=0 1 2n+1 +µ(D)ΨA 12.4NΨ−11 h ≤1 2+µ(D)Ψ16A.h2 3h2 0 Ψ−11 h, b eause 1 4N≤8h h02 . We an hoose h1 small enough to have: µ(D)Ψ16A.h2 3h2 0 Ψ−11 h≤1 2 for every h∈(0, h1) , sine lim h→0h2Ψ−11 h= 0 . We get for suh hZD ΨM|uξ,1−h|dµ ≤1, so that kuξ,1−hkLΨ(µ)≤1 M=24 A 1 Ψ−1(1/h) , as it was announed.  Examples and ounterexamples. We are going to give some examples showing that we do not have the reverse impliations in general in Theorem 4.10 and Theorem 4.11. 1. Condition ( R ) is not suient in general to have a ontinuous emb edding. Let Ψ(x) = ex−1 (note that this Orliz funtion even fullls the ∆1 ondition !). Note that Ψ(AΨ−1(1/h)) ∼h−A , when h→0 . a. Let ν b e a probability measure on T , supp orted by a ompat set L of Leb esgue measure zero, suh that ν(I)≤ |I|1/2 , for eah I . We an asso iate to ν the measure on D dened by ˜ν(E) = ν(E∩T) . Then the identity map from HΨ to LΨ(˜ν) is not even dened. Nevertheless the ondition ( R ) is learly fullled with A= 1/2 . 53 b. Now, we exhibit a similar example (less artiial) on the op en disk. Let ν b e as previously. By a standard argument: for every integer n , there exists a funtion gn in the unit ball of the disk algebra suh that |gn|= 1 on L and kgnkΨ≤4−n . As L is ompat, there exists some rn∈(1/2,1) suh that |gn(rnz)| ≥ 1/2 for every z∈L . Now, dene the measure µ by: µ(E) = ∞ X n=1 1 2nνn(E), where: νn(E) = ν{z∈T|rnz∈E}. If W is a Carleson window of size h then, for eah n≥1 , we have: ν{z∈T|rnz∈W}≤ν(W∩T)≤(2h)1/2. Hene, µ(W)≤(2h)1/2 and the ondition ( R ) is fullled. Nevertheless, the identity from HΨ to L1(µ) is not ontinuous: kgnkΨ≤4−n ; but kgnk1≥1 2nZrnT|gn|dνn≥1 2nZL|gn(rnw)|dν(w)≥1 2n+1 · 2. Condition ( K ) is not neessary in general to have a ontinuous embedding. When Ψ satises ∆2 , the identity from HΨ to LΨ(µ) is ontinuous if and only if µ is a Carleson measure. So the onditions ( R ), ( K ) and the ontinuity are equivalent in this ase. Atually, when Ψ do es not satisfy ∆2 , we onstrut b elow a measure µ on D suh that the identity from HΨ to LΨ(µ) is ontinuous and order b ounded , but µ is not a Carleson measure ( a fortiori do es not verify ( K )). Note that the measure µ is then Ψ -Carleson but not Carleson. Here is the example: We have assumed that Ψ do es not satisfy ∆2 ; so there exists an inreasing sequene (an)n≥1 suh that Ψ(an) n is inreasing and Ψ(2an) Ψ(an)≥n2n . Now, dene the disrete measure µ=∞ X n=1 n Ψ(2an)−n+ 1 Ψ(2an+1)δxn, where: xn= 1 −1 Ψ(2an)· As µ[xN,1]=N Ψ(2aN) , the measure µ is not Carleson: it should b e b ounded by c(1 −xN) = c Ψ(2aN) , where c is some onstant. 54 We know that for every f in the unit ball of HΨ and every x∈(0,1) , we have |f(x)| ≤ 4Ψ−11 1−x : see Lemma 3.11. So we only have to see that g∈LΨ(µ) , where g(x) = Ψ−11 1−x . Indeed, we have: ZD Ψ|g| 2dµ =∞ X n=1 n Ψ(2an)−n+ 1 Ψ(2an+1)Ψ|g(xn)| 2 ≤∞ X n=1 n Ψ(2an)Ψ1 2Ψ−11 1−xn ≤∞ X n=1 nΨ(an) Ψ(2an)≤∞ X n=1 1 2n ≤1 so kgkΨ≤2 . 3. Condition ( K0 ) is not neessary in general to have a ompat emb edding. We an nd, for every Orliz funtion Ψ not satisfying ∇0 , a measure µ suh that the identity from HΨ to LΨ(µ) is ompat but ( K0 ) is not satised. Indeed: sine Ψ/∈ ∇0 , we an selet two inreasing sequenes (xn)n≥1 and (yn)n≥1 , with 1≤xn≤yn≤xn+1 and Ψ(xn)>1 , and suh that lim xn= +∞ and: Ψ(2xn) Ψ(xn)≥Ψ(2nyn) Ψ(yn)· Dene rn= 1 −1 Ψ(yn) and the disrete measure: µ=∞ X n=1 1 Ψ(2nyn)δrn. The series onverge sine Ψ(2nyn)≥2n . Thanks to Lemma 3.11, we have, for every f in the unit ball of HΨ and every n≥1 : |f(rn)| ≤ 4Ψ−11 1−rn= 4yn. Given r > r1 , there exists an integer N≥1 suh that rN< r ≤rN+1 . Then, 55 for every f in the unit ball of HΨ , we have kfkLΨ(D\rD,µ)≤2−N+2 , sine: ZD\rD Ψ|f| 2−N+2 dµ =∞ X n=N+1 1 Ψ(2nyn)Ψ|f(rn)| 2−N+2  ≤∞ X n=N+1 Ψ(2Nyn) Ψ(2nyn) ≤∞ X n=N+1 1 2n−N= 1. This implies that lim r→1sup kfkΨ≤1kfkLΨ(D\rD,µ)= 0 . By Prop osition 4.9, the identity from HΨ to LΨ(µ) is ompat. On the other hand, writing hn=1 Ψ(xn) and tn=1 Ψ(yn) , we have: Kµ(hn)≥µ([1 −tn,1]) tn = Ψ(yn)∞ X m=n 1 Ψ(2mym)≥Ψ(yn) Ψ(2nyn) ≥Ψ(xn) Ψ(2xn)=1/hn Ψ2Ψ−1(1/hn) , and this shows that ( K0 ) is not satised. 4. Condition ( R0 ) is not suient in general to have a ompat embedding. We an nd an Orliz funtion Ψ and a vanishing Ψ -Carleson measure ( i.e. ( K0 ) is satised) µ suh that the identity from HΨ to LΨ(µ) is not ompat. We shall use the Orliz funtion introdued in [25℄. The key prop erties of this funtion Ψ are: i) For every x > 0 , Ψ(x)≥x3/3 . ii) For every integer k≥1 , Ψ(k!) ≤(k!)3 . iii) For every integer k≥1 , Ψ(3(k!)) > k.(k!)3 . One again, the job is done by a disrete measure. Dene xk=k! ; yk=(k+ 1)! k1/3 ; rk= 1 −1 Ψ(yk) and ρk= 1 −1 Ψ(xk)· Of ourse, x2< y2< x3<··· . Let ν b e the disrete measure dened by: ν=∞ X k=2 νk, where: νk=1 Ψ(k+ 1)!X ak2=1 δrka. 56 Observe that kνkk ≤ k2 Ψ(k+ 1)!≤3.k2 (k+ 1)!3 so that the series onverges. Note that ν is supp orted in the union of the irles of radii rk and not in a subset of the segment [0,1] as in the preeding ounterexamples. In order to show that ( R0 ) is satised, it is learly suient to prove that, when 1 Ψ(yk)≤h < 1 Ψ(yk−1) (with k≥3 ), we have: ρν(h)≤1 Ψk1/3 2Ψ−1(1/h)· Supp osing then 1 Ψ(yk)≤h < 1 Ψ(yk−1) , we have Ψ−1(1/h)≤yk so Ψk1/3 2Ψ−1(1/h)≤1 2Ψ(k+ 1)!. Therefore, it is suient to establish that ρν(h)≤2 Ψ(k+ 1)!· A Carleson window W(ξ, h) (where ξ∈T ) an ontain at most one k2 -ro ot of the unity, sine 2h < 2 Ψ(yk−1)≤6 y3 k−1≤6(k−1) (k!)3≤2π k2· This implies that νkW(ξ, h)≤1 Ψ(k+ 1)!· Nevertheless, when j < k , the window W(ξ, h) annot meet any irle of radius rj (entered at the origin), so νjW(ξ, h)= 0 . We obtain: νW(ξ, h)=∞ X j=k νjW(ξ, h)≤1 Ψ(k+ 1)!+X j>k j2 Ψ(j+ 1)! ≤1 Ψ(k+ 1)!+∞ X j=k+1 3j2 (j+ 1)!3 ≤1 Ψ(k+ 1)!+3 (k+ 1)!3 ∞ X s=1 1 k+ 1s ≤2 Ψ(k+ 1)!· This proves that ( R0 ) is satised. Let us intro due the funtion fk(z) = xku1,ρkzk2=xk1−ρk 1−ρkzk22 . It lies in the unit ball of HΨ by Corollary 3.10: 57 Let J b e the ar: J={eit ;−ρ < t < ρ}. The map ψ is regular on J , and so there exist two onstants γ1 and γ2>0 suh that for every Borel subset E of J , one has: γ1m(E)≤mψ(E)≤γ2m(E). If now β∈(0,1) , we put: ψβ(z) = (ψ(z))β. Then, it is easy to see that for every Borel subset E of J , one has: mψβ(E)=β mψ(E), and so γ1β m(E)≤mψβ(E)≤γ2β m(E). In order to prove (4.20) , onsider the funtion F:D→C dened by: F(z) = feiδψβ(z), where β= 4δ/π . Then: Re F(0)=u(1 −√2)βeiδ≤Nu(eiδ)≤α. Let us all χ the map: χ(z) = eiδψβ(z). It is lear that I+ is ontained in χ(J) . If A={|f∗|> λ}∩I+ , then E=χ−1(A) is a Borel subset of J , and for every ξ∈E , one has |F∗(ξ)|> λ . Then: m(A) = mχ(E)=mψβ(E)≤γ2β m(E) ≤8γ2 δ 2πm({|F∗|> λ}) = 8γ2m(I+)m({|F∗|> λ}) and using Lemma 4.24 for F , ≤8γ2m(I+)×8c3Re F(0) λ3 ≤64γ2c3α λ3m(I+) = k2α λ3m(I+). The pro of of (4.20) is nished, and Prop osition 4.25 follows.  64 4.4.3 Pro of of Theorem 4.19 Using the fat h7→ µφS(ξ, h) is nondereasing, it is enough to prove that there exist h0>0 , and ε0>0 , suh that (4.15) is true for 0< h < h0(1−|φ(0)|) , and 0< ε < ε0 , b eause hanging the onstant k1 , if neessary, the theorem will follow. We an also supp ose that ξ= 1 . The real part of 1/1−φ(z) is positive, in fat greater than 1/2, for every z∈D . Take 0< h < h0 , and onsider the analyti funtion f dened by: f(z) = h 1−φ(z)1/3 , where the ubi root is taken in order that, for every z∈D , f(z) b elongs to the one C={z∈C:−π/6<Arg (z)< π/6}. Clearly µφS(1, h)=m({|f∗|>1}) and µφS(1, εh)=m({|f∗|>1/3 √ε}) . We also have: |f(0)| ≤ h 1−|φ(0)|1/3 <3 ph0. We shall write f=u+iv where u and v are real-valued harmoni funtions. Observe that: |v(z)|<1 √3u(z), for every z∈D . It is known that f∈Hp , for every p < 3 (see [12℄, Theorem 3.2), and so u and v are the Poisson integrals of u∗ and v∗ (in partiular u, v ∈h1 ). We are lo oking for a ontrol of m({|f∗|>1/3 √ε}) by ε times m({|f∗|>1}) . Prop osition 4.26 provides this ontrol replaing m({|f∗|>1}) by m({Nu > 2}) : m({|f∗|>1/3 √ε})≤8k2εm({Nu > 2}), when m({Nu > 2})<1/4 . As f is valued in C , |f∗| is ontrolled by u∗ . Then what we need in fat is a ontrol of the measure of level sets of Nu by the measure of level sets of u∗ . This will b e done by using Prop osition 4.27. Indeed, by Prop osition 4.27, we have: m({Nu > 2})≤m({Mu∗>2})≤k4m({u∗>1}). We know that ||u∗||1=u(0) ≤ |f(0)| ≤ h1/3 0 . Then, hoosing h0 small enough, we an have k4h1/3 0<1/4 , and so m({Nu > 2})<1/4 ; therefore, we an use Prop osition 4.26. Moreover we also have {u∗>1} ⊆ {|f∗|>1} . Taking ε0<3√3/64 , we have, for 0< ε < ε0 , by Prop osition 4.26: m({|f∗|>1/3 √ε})≤8k2ε m({Nu > 2})≤8k4k2ε m({u∗>1}) ≤k1ε m({|f∗|>1}), taking k1= 8k4k2 .  65 5 Bergman spaes 5.1 Bergman-Orliz spaes Denition 5.1 Let dA(z) = dx dy π ( z=x+iy ) be the normalized Lebesgue measure on D . The Bergman-Orliz spae BΨ denotes the spae of analyti funtions f:D→C whih are in the Orliz spae LΨ(D, dA) . The Bergman- Morse-Transue spae is the subspae BMΨ=BΨ∩MΨ(D, dA). BΨ , equipp ed with the indued norm of LΨ(D, dA) , is a Banah spae, as an obvious onsequene of the following lemma, analogous to Lemma 3.11. Lemma 5.2 For every a∈D , the norm of the evaluation funtional δa , whih maps f∈BΨ to f(a) , is: kδak ≈ Ψ−11 (1 −|a|)2· Pro of. For every analyti funtion g:D→C , the mean-value prop erty gives: g(0) = ZD g(z)dA(z). Hene if φa:D→D denotes the analyti automorphism φa(z) = z−a 1−¯az , whose inverse is φ−1 a=φ−a , one has, for every f∈BΨ , using the hange of variable formula: f(a) = f◦φ−a(0) = ZD f◦φ−a(z)dA(z) = ZD f(w)|φ′ a(w)|2dA(w) =ZD f(w)Ha(w)dA(w), where: Ha(w) = |φ′ a(w)|2=(1 −|a|2)2 |1−¯aw|4· The kernel Ha plays for BΨ the role that the Poisson kernel Pa plays for HΨ : the analyti reproduing kernel Ka for B2 b eing Ka(z) = 1 (1−¯az)2· We therefore have (using [30℄, Prop osition 4, page 61, or [5℄, Theorem 8.14): |f(a)| ≤ 2kfkΨkHakΦ, whih proves the ontinuity of δa . To estimate its norm, we are going to ma jorize kHakΦ , with the help of Lemma 3.9. Let us notie that, on the one hand, kHak1= 1 (take f= 1I in the ab ove identity); and on the other hand: kHak∞=(1 −|a|2)2 (1 −|a|)4=(1 + |a|)2 (1 −|a|)2· 66 We get, setting b=kHak∞ , and using Lemma 3.9 for k kΦ : kHakΦ≤b Φ−1(b)· But b≤Φ−1(b)Ψ−1(b) (see [30℄, Prop osition 1 (ii), page 14). Hene kHakΦ≤ Ψ−1(b) . Now: b≤4 (1 −|a|)2· We have Ψ−1(4t)≤4Ψ−1(t) for all t > 0 . It follows that kHakΦ≤CΨ−11 (1 −|a|)2· Sine kδak ≤ 2kHakΦ , we get the upp er b ound in Lemma 5.2. For the lower bound, we simply observe that Ha=|Ga| , where Ga(z) = (1 −|a|2)2 (1 −az)4 , and by Lemma 3.9: kδak ≥ |Ga(a)| kGakΨ =|Ha(a)| kHakΨ≥ 1 (1 −|a|2)2 b/Ψ−1(b)=Ψ−1(b) b(1 −|a|2)2 =Ψ−1(b) (1 + |a|)4≥1 16Ψ−1(b)≥1 16Ψ−11 (1 −|a|)2, sine b≥1/(1 −|a|)2 .  Prop osition 5.3 We have the fol lowing properties i) BMΨ is the losure of H∞(D) in LΨ(D,A) and atual ly the algebrai polynomials are dense in BMΨ . ii) On the unit bal l of BΨ , the weak-star topology σLΨ(D,A), MΦ(D,A) oinides with the topology of onvergene on ompat subsets of D . iii) BΨ is losed in LΨ(D,A) for the weak-star topology. iv) If Ψ∈ ∇2 , BΨ is (isometri to) the bidual of BMΨ . Pro of. For the rst p oint, let us x f∈BMΨ . Setting fr(z) = f(rz) for z∈D and 0≤r < 1 , it sues to show that kfr−fkΨ−→ r→10 , sine, b eing analyti in the disk rD⊂D , fr an b e uniformly approximated on D by its Taylor series. But the norm of MΨ is absolutely ontinuous (see [30℄, Theorem 14, page 84) and therefore, for every ε > 0 , there is some R > 0 , with 1/3≤R < 1 , suh that kf1ID\RDkΨ≤ε ; hene: ZD\RD Ψ|f| 4εdA≤ZD\RD 1 4Ψ|f| εdA≤1 4· 67 When r≥2R R+1 ≥1/2 , we therefore have: ZD\1+R 2D Ψ|fr| 4εdA≤1, and, by onvexity of Ψ : ZD Ψ|fr−f| 8εdA≤Z1+R 2D Ψ|fr−f| 8εdA +ZD\1+R 2D 1 2hΨ|fr| 4ε+ Ψ|f| 4εidA ≤Z1+R 2D Ψ|fr−f| 8εdA +1 2ZD\1+R 2D Ψ|fr| 4εdA+1 2ZD\RD Ψ|f| 4εdA ≤1, for r lose enough to 1 sine fr−f tends to 0 uniformly on 1+R 2D . Hene, for some r0<1 , one has kfr−fkΨ≤8ε for r0≤r < 1 . This was the laim. ii) It sues to use a sequential argument sine the top ologies are metrizable on balls (the spae MΦ(D,A) is separable). Assume that f∈BΨ (with kfkΨ≤ 1 ) is the weak-star limit of a sequene of analyti funtions fn∈BΨ (with kfnkΨ≤1 ). Testing this with the funtion hk(z) = (k+ 1)zk , we obtain that the kth Taylor o eient ak(n) of fn onverges to the kth Taylor o eient ak of f : ak(n) = ZD fnhkdA−→ ZD fhkdA=ak, for every k≥0. Fix a ompat K⊂D , there exists an r∈(0,1) suh that K⊂rD . We have: sup z∈K|fn(z)−f(z)| ≤ X k≥0|ak(n)−ak|rk−→ 0 by the dominated onvergene Theorem (observe that |ak−ak(n)| ≤ 2(k+ 1) for every k≥0 and every n≥0 ). For the onverse, supp ose now that fn∈BΨ (with kfnkΨ≤1 ) onverges uniformly on every ompat subsets of D to f∈BΨ (with kfkΨ≤1 ). Fixing g∈MΦ(D,A) and ε > 0 , there exists an r∈(0,1) suh that kg1ID\rDkΦ≤ε . We have: ZD (fn−f)gdA≤ZrD (fn−f)gdA+ 2ε≤sup z∈rD|fn(z)−f(z)|.kgk1+ 2ε. By hyp othesis sup z∈rD|fn(z)−f(z)| −→ 0 . The onlusion follows. 68 iii) By the lassial Theorem of Banah-Dieudonné, it is suient to prove that the balls are weak-star losed (equivalently weak-star ompat) and by separability of MΦ(D,A) , the weak-star top ology is metrizable on balls. The previous fat ii) shows that it is equivalent to prove that the unit ball of BΨ is ompat for the top ology of onvergene on ompat subsets. But this is easy: indeed, if fn in the unit ball of BΨ . This is a normal family thanks to Lemma 5.2. A subsequene onverges to an analyti funtion f on ompat subsets of D and the Fatou Lemma implies that f atually lies in the unit ball. iv) Assume now that Ψ satises ∇2 . Sine (MΨ)∗∗ =LΨ(D,A) , we have (BMΨ)∗∗ =BMΨw∗ , in the spae LΨ(D,A) . Hene it sues to show that BMΨw∗ =BΨ . We already know that BΨ is weak-star losed. Now, let f∈BΨ . Obviously, fr∈BMΨ for every r∈(0,1) , where fr(z) = f(rz) . Moreover kfrkΨ≤ kfkΨ . But this is lear that fr is uniformly onvergent to f on ompat subsets of D , when r→1− . By ii), the onlusion follows.  In the previous pro of, we an see the p oints ii) and iii) in a slighlty dierent way: the unit ball B of BΨ is ompat for the top ology τ of uniform onvergene on ompat subsets. Then observe that the identity from B , equipp ed with the top ology τ , to B , equipp ed with the weak-star top ology, is ontinuous (the sequential argument is suient by metrizability). This implies, sine the weakstar top ology is separated, that B is weak-star ompat (hene losed) and that the top ologies oinide. Now, by Banah-Dieudonné, the spae BΨ is weak-star losed. 5.2 Compat omp osition op erators on Bergman-Orliz spaes We shall b egin with showing that, as in the Hardy-Orliz ase, every symb ol φ denes a b ounded omp osition op erator. Prop osition 5.4 Every analyti self-map φ:D→D indues a bounded omposition operator Cφ:BΨ→BΨ . Moreover, Cφf∈BMΨ for every f∈BMΨ ; hene, when Ψ∈ ∇2 , the former operator is the bi-transposed of Cφ:BMΨ→ BMΨ . Pro of. It sues to follow the lines of Prop osition 3.12, and to integrate the integrals written there b etween 0 and 1 , with resp et to the measure 2rdr .  Before stating and proving the main theorem, we are going to prove the following auxiliary result, interesting in itself, and whih reinfores an example of J. H. Shapiro ([36℄, Example, page 185). Prop osition 5.5 There exists a Blashke produt B having angular derivative at no point of T=∂D , in the fol lowing sense: (5.1) (∀ε > 0) (∃cε>0) 1 −|B(z)| ≥ cε(1 −|z|)ε,∀z∈D. 69 Pro of. We shall take: B(z) = +∞ Y n=1 |zn| zn zn−z 1−¯znz , where: {zn;n≥1}=[ n≥1 An, with: An={rnωj n;ωn= e2πi/pn,0≤j≤pn−1}, where (rn)n≥1 is a (stritly) inreasing sequene with 0< rn<1 , and the integers pn will have to be adjusted, satisfying the Blashke ondition: +∞ X n=1 (1 −|zn|) = +∞ X n=1 pn(1 −rn)<+∞. One has: |B(z)|2= +∞ Y n=1 zn−z 1−¯znz 2= +∞ Y n=1 1−(1 −|z|2)(1 −|zn|2) |1−¯znz|2≤exp −S(z), where: S(z) = +∞ X n=1 (1 −|z|2)(1 −|zn|2) |1−¯znz|2· We now proeed to minorize S(z) . For this purpose, we shall need the following simple lemma, whose pro of will b e temp orarily p ostp oned. Lemma 5.6 For every positive integer p and every a∈D , one has, setting ω= e2πi/p : 1 p p−1 X k=0 1 |1−aωk|2=1−|a|2p 1−|a|2 1 |1−ap|2≥1 4p|a|p. Then, setting r=|z| , we have: S(z)≥(1 −r) +∞ X n=1 (1 −rn) pn−1 X k=0 1 |1−rnω−k nz|2 ≥1−r 4 +∞ X n=1 p2 n(1 −rn)(rnr)pn. We shall take: pn=(1 −rn)εn−1+ 1 , with: εn=1 √n and rn= 1 −1 2n 70 and where [ ] stands for the integer part. More expliitly: pn=2n−√n+ 1. If r≥1/2 , let N≥1 b e suh that rN< r ≤rN+1 . One has: S(z)≥1−r 4(1 −rN)2εN−1r2pN N≥1 8(1 −rN)2εNr2pN N, sine 1−r≥1−rN+1 = (1 −rN)/2 . Moreover: pN≤2(1 −rN)εN−1= 2.2N(1−εN)≤2.2N, so that: S(z)≥1 8(1 −r)2εN(1 −2−N)2N+1 ≥c(1 −r)2εN, where c is a p ositive numerial onstant. Hene, setting: ε(z) = 2εN for |z| ≥ 1/2 and rN< r ≤rN+1 , one has: ε(z)−→ |z|< →1 0, and we get: 1−|B(z)|2≥1−e−S(z)≥1−e−c(1−|z|)ε(z)≥c′(1 −|z|)ε(z), where c′ is another p ositive numerial onstant. This gives ondition (5.1), sine 1−|B(z)| ≥ 1−|B(z)|2 2· Finally, the Blashke ondition is satised, sine: pn(1 −rn)≤2(1 −rn)εn= 2.2−√n. This ends the pro of of Prop osition 5.5.  Pro of of Lemma 5.6. Let Gp b e the nite group of pth ro ots of unity, equipp ed with its normalized Haar measure. For u:Gp→C and 0≤k≤p−1 , we denote by ˆu(k) the kth Fourier o eient of u , i.e. : ˆu(k) = 1 pX z∈Gp u(z)z−k. Then, the Planherel-Parseval formula for Gp reads: p−1 X k=0 |ˆu(k)|2=1 pX z∈Gp|u(z)|2. 71 Applying this to u(z) = 1 1−az =X l≥0 0≤k≤p−1 alp+kzk= p−1 X k=0 ˆu(k)zk, with ˆu(k) = X l≥0 alp+k=ak 1−ap , we get: 1 p p−1 X k=0 1 |1−aωk|2= p−1 X k=0 |a|2k |1−ap|2=1−|a|2p (1 −|a|2)|1−ap|2· To nish, we note that |1−ap| ≤ 2 , and that, by the arithmetio-geometri inequality, we have, with x=|a|2 : 1−|a|2p 1−|a|2= 1 + x+···+xp−1 ≥px1+2+···+(p−1)1/p =p xp−1 2≥p xp/2=p|a|p.  Theorem 5.7 If the omposition operator Cφ:BΨ→BΨ is ompat, then (5.2) Ψ−11 (1 −|φ(a)|)2 Ψ−11 (1 −|a|)2−→ |a|< →1 0. This ondition is suient if Ψ∈∆2 . Before giving the pro of of this theorem, let us note that in ase where Ψ = Ψ2 , with Ψ2(x) = ex2−1 , it reads: the omposition operator Cφ:BΨ2→BΨ2 is ompat if and only if: (5.3) (∀ε > 0) (∃cε>0) 1 −|φ(z)| ≥ cε(1 −|z|)ε,∀z∈D. Indeed, Ψ2∈∆2 , and: Ψ−1 21 (1 −|φ(a)|)2 Ψ−1 21 (1 −|a|)2=rlog 1 (1 −|φ(a)|)2 rlog 1 (1 −|a|)2 ; hene Cφ is ompat if and only if log 1 (1 −|φ(a)|)2 log 1 (1 −|a|)2 −→ |a|→10. 72 Then, for every ε > 0 , we an nd some cε>0 suh that, for all a∈D : log 1 1−|φ(a)|≤εlog 1 (1 −|a|)2+cε; whih is equivalent to (5.3).  That allows to have ompat omposition op erators on BΨ2 whih are not ompat on HΨ2 . However it is very likely that this is the ase for every Ψ∈∆2 , but we have not tried to see this full generality. Theorem 5.8 There exist symbols φ:D→D suh that the omposition operators Cφ is ompat from BΨ2 into itself, but not ompat from HΨ2 into itself, and even Cφ is an isometry onto its image. A similar example is well-known for the Hilb ert spaes B2 and H2 (see [36℄, pages 180186). Pro of. Let B b e a Blashke pro dut verifying the ondition of Prop osition 5.5. We intro due φ(z) = zB(z) . The funtion still veries 1−|φ(z)| ≥ Cε(1 −|z|)ε , sine |φ(z)| ≤ |B(z)| on D . From Theorem 5.7, it follows, sine Ψ2∈∆2 , that Cφ:BΨ2→BΨ2 is ompat. We are now going to see that Cφ:HΨ2→HΨ2 is an isometry. Indeed, reall the following well-known fat, that we already used (see [29℄, Theorem 1): sine φ is an inner funtion, the image φ(m) of the Haar measure m of T under φ is equal to Pa.m , where a=φ(0) and Pa is the Poisson kernel at a . Here φ(0) = 0 so φ(m) = m . It follows that for every f∈HΨ2 , one has, for C > 0 : ZT Ψ2|f◦φ| Cdm =ZT Ψ2|f| Cdm so that kfkΨ2=kf◦φkΨ2 .  We shall need also the following lemma, whih ompletes Lemma 5.2. Lemma 5.9 For every f∈BMΨ , one has: f(a) = oΨ−11 (1 −|a|)2 as |a|< −→1. Pro of. This is obvious for the monomials en:z7→ zn sine |en(a)| ≤ 1 , whereas Ψ−11/(1 −|a|)2−→ |a|→1+∞ . Sine the evaluation δa is b ounded on BMΨ and δa/Ψ−1(1/1−|a|2)=O(1) , it sues to use that the p olynomials are dense in BMΨ ; but this was already proved in Prop osition 5.3.  Pro of of Theorem 5.7. If Cφ:BΨ→BΨ is ompat, then so is the restrition Cφ:BMΨ→BMΨ and its adjoint C∗ φ=Cφ:BMΨ∗→BMΨ∗ . Sine 73