Monsters in Hardy and Bergman spaces

Mons e s in Ha dy and Be gman spaces L. BERNAL–GONZ´ ALEZ and M.C. CALDER´ ON–MORENO Depa amen o de An´alisis Ma em´a ico, Facul ad de Ma em´a icas, Apdo. 1160, A da. Reina Me cedes, 41080 Se illa, Spain. E–mails: lb[email p o ec ed], [email p o ec ed] Abs ac A mons e in he sense o Luh is a holomo phic unc ion on a simply con- nec ed domain in he complex plane such ha i and all i s de i a i es and an ide i a i es exhibi an ex emely wild beha iou nea he bounda y. In his pape he Ha dy spaces Hpand he Be gman spaces Bp(1 ≤p < ∞) on he uni disk a e conside ed, and i is shown ha he e a e no Luh-mons e s in hem. Ne e heless, i is p o ed ha T-mons e s (as in oduced by he au ho s in an ea lie wo k) can be ound in each o hese spaces o any ini e o de linea di e en ial ope a o T. Key wo ds and ph ases: Luh-mons e , T-mons e , Ha dy space, Be gman space, s ongly omnip esen ope a o , di e en ial ope a o , hype cyclic unc- ion. 2000 Ma hema ics Subjec Classi ica ion: P ima y 30E10. Seconda y 30D40, 30H05, 47A16, 47B38. 1 In oduc ion As soon as he exis ence o a ma hema ical en i y is es ablished, a na u al p oblem a ises: Do such en i ies exis wi h addi ional (e en “mo e pe ec ”) p ope ies? This 0 has been he line o esea ch which has mo i a ed his pape , his ime in he se ing o holomo phic unc ions wi h “wild” beha iou in he bounda y o he open uni disk D={|z|<1}. Suppose ha Gis a domain in he complex plane C; by H(G) we deno e as usual he F ´eche space o all holomo phic unc ions on G, endowed wi h he compac -open opology. I Gis simply connec ed hen i has been p o ed in 1985 by W. Luh [21] he exis ence o a dense se o unc ions –which he called “mons e s”– in H(G) such ha all i s de i a i es and an ide i a i es exhibi an ex emely wild beha iou nea he bounda y o G. Such a chao ic p ope y can be exp essed in e ms o ce ain gene alized clus e se s, in oduced by Luh himsel . In 1987 G osse-E dmann [18], see also [19, Sec ion 4.b], showed ha , in ac , he e is a esidual se o mons e s. Fu he in e es ing esul s on his opic can be seen in [22–24]. Wi h he aim o inding ope a o s which a e di e en om hose o di e en ia- ion and an idi e en ia ion unde whose ac ion he e a e holomo phic unc ions wi h bounda y wild beha iou , he au ho s ha e ecen ly in oduced [5] he no ions o T-mons e s and s ongly omnip esen ope a o s, see De ini ion 1.1 below. Le us ix some e minology and no a ion. By ∂G we deno e he bounda y o a domain G⊂Cin he ex ended complex plane C∞=C∪ {∞}.Nis he se o posi i e in ege s, N0=N∪ {0}and Ris he eal line. An ope a o always e e s o a con inuous (no necessa ily linea ) sel mapping. We deno e by O(∂G) he se o all open subse s o C∞mee ing he bounda y o G. I A⊂C hen A ep esen s he closu e o A,k kA:= supz∈A| (z)|, whe e is a complex unc ion de ined in A, and LT(A) is he se o all a ine linea ans o ma ions τ,τ(z) = az +b, such ha τ(D)⊂A. In he ollowing de ini ion, we a e allowing he poin o in ini y o be a bounda y poin o Gwhen Gis unbounded (as in [7, 9, 10]). Obse e also ha he domain Gis allowed o be non-simply connec ed in (a)–(c). De ini ion 1.1. (a) A unc ion ∈H(G) is a holomo phic mons e whene e he ollowing uni e sali y p ope y is sa is ied: Fo each g∈H(D) and each ∈∂G he e exis s a sequence (τn) o a ine linea ans o ma ions wi h τn(z)→ (n→ ∞) uni o mly on Dand τn(D)⊂G(n∈N) such ha (τn(z)) →g(z) (n→ ∞) locally uni o mly in D. 1 (b) Le T:H(G)→H(G) be an ope a o . Then a unc ion ∈H(G) is a T– mons e i T is a holomo phic mons e . The se o T–mons e s is deno ed by M(T). (c) An ope a o T:H(G)→H(G) is s ongly omnip esen i o all g∈H(D), ε > 0, ∈(0,1) and V∈O(∂G) he se U(T, g, ε, , V ) := { ∈H(G) : he e exis s some τ∈LT(V∩G) such ha k(T )◦τ−gk D< ε} is dense in H(G). (d) I Gis simply connec ed, hen a unc ion ∈H(G) is a Luh–mons e whene e e e y de i a i e (n)(n∈N0) and e e y an ide i a i e (−n)(n∈N) is a holomo phic mons e . Obse e ha is a holomo phic mons e i and only i i is an I-mons e (I:= he iden i y ope a o ), and ha is a Luh-mons e i and only i is simul aneously aDN-mons e ( o all N∈N0) and a D−N a-mons e ( o all N∈Nand all a∈G). He e DN(N∈N0) is he di e en ia ion ope a o DN = (N),D0=I, and D−N a is he an idi e en ia ion ope a o gi en by D−N a := he unique an ide i a i e Fo o de No such ha F(a) = · · · =F(N−1)(a) = 0. I happens ha an ope a o Ton H(G) is s ongly omnip esen i and only i he se M(T) is esidual (see [5, Theo em 2.2]). Hence G osse-E dmann [18, Kapi el 3] had showed in ac ha e e y DNand e e y D−N ais s ongly omnip esen . He and he au ho s ha e iden i ied se e al kinds o s ongly omnip esen ope a o s, including in- ini e o de di e en ial and an idi e en ial ope a o s, in eg al ope a o s, composi ion and mul iplica ion ope a o s [5, 9, 10]. Fo o he kinds o ope a o s –also in oduced by he au ho s– unde whose ac ion ce ain unc ions ha e some ype o bounda y chao ic beha iou , he eade is e e ed o [1] (omnip esen ope a o s), [2, 6, 14] (DI-ope a o s) and [7] ( o ally omnip esen ope a o s), see also [8]. I happens ha e e y o ally omnip esen ope a o is s ongly omnip esen and DI, and ha i an ope a o is ei he s ongly omnip esen o DI hen i is omnip esen . By using o ally omnip esen ope a o s, he au ho s ha e ecen ly 2 p o ed (see [7]) ha he e is a dense linea mani old in H(G) all o whose nonze o unc ions a e Luh-mons e s. We will need some e minology abou uni e sali y (see [19] o an excellen su ey, upda ed ill 1999). I Xand Ya e (Hausdo ) opological ec o spaces o e he same ield K(= Ro C) and Tn:X→Y(n∈N) is a sequence o con inuous linea mappings, hen (Tn) is said o be hype cyclic (o uni e sal) i and only i he e is a ec o x∈X, called also hype cyclic o (Tn), such ha he o bi {Tnx:n∈N} is dense in Y. The sequence (Tn) is called densely hype cyclic whene e he se HC((Tn)) o hype cyclic ec o s o (Tn) is dense. I X=Yand Tis a linea ope a o on X hen Tis called hype cyclic i and only i he sequence (Tn) o i e a es is hype cyclic. I is easy o see ha in such a case (Tn) is indeed densely hype cyclic. The exis ence o T-in a ian dense linea mani olds o hype cyclic ec o s o each hype cyclic linea ope a o Ton a ( eal o complex) locally con ex space was shown by He e o, Bou don and B`es [11, 12, 20] (see also [4] o he addi ional p ope y o maximal algeb aical ca dinali y o such mani olds). In 1999 he i s au ho ex ended his esul o hype cyclic sequences o mappings, see [3] and Theo em 2.6. Fo he sake o con enience, we will keep in his wo k he no ion o hype cyclici y e en when he spaces X, Y and he mappings T, Tna e no linea . The aim o his pape is o s udy he exis ence o Luh-mons e s and in gene al o T-mons e s in (no necessa ily closed) subspaces o H(G), mainly in he maybe mos emblema ic spaces o analy ic unc ions on G=D, namely, he Ha dy spaces Hpand he Be gman spaces Bp(1 ≤p < ∞). Recall ha Hpis he class o all unc ions ∈H(D) sa is ying k kp:= sup 0< <1Z2π 0 | ( eiθ)|pdθ 2π1/p <∞, while Bpis he class o all ∈H(D) sa is ying k kp:= ZD | (z)|pdA(z)1/p <∞. Each one becomes a Banach space unde he co esponding no m k · kp. Recall also ha Hp⊂Bpwi h con inuous inclusion. We ha e deno ed by dA(z) he a ea measu e on Dno malized so ha he a ea o Dis 1. The exis ence o dense linea mani olds o holomo phic mons e s in hese spaces is also conside ed. 3 2 Mons e s in subspaces o H(D) Up o da e, all esul s abou mons e s ha e been add essed o s a e hei exis ence, wi h success excep o e y special examples. Howe e , in he cu en se ing, we a e on he poin o ob ain a gene al s a emen o non-exis ence o classical mons e s in unc ion spaces, see Theo em 2.2 below. Be o e his, we es ablish an auxilia y lemma whose con en is p obably well known. Since we ha e no been able o ind a e e ence o i , we p o ide wi h an elemen a y p oo . Lemma 2.1. I ∈H(D)and 0∈Bp o some p∈[1,∞) hen ∈Hp. P oo . By hypo hesis, RD| 0(z)|pdA(z)<∞. Passing o pola coo dina es, we ge R1 0R2π 0| 0(seiθ)|ps dsdθ < ∞. Since | 0|pis Lebesgue-in eg able on a neighbou hood o he o igin, we can d op he ac o s, ha is, Z1 0Z2π 0 | 0(seiθ)|pdsdθ < ∞. I is e iden ha we can suppose (0) = 0. Then ( eiθ) = R 0 0(seiθ)eiθ ds o all ∈[0,1) and all θ∈[0,2π]. Hence, by he H¨olde inequali y, | ( eiθ)|p≤Z 0 | 0(seiθ)|dsp ≤ p−1Z 0 | 0(seiθ)|pds ≤Z1 0 | 0(seiθ)|pds. Disca ding he e ms in he middle, an in eg a ion o e [0,2π] yields he desi ed esul . Theo em 2.2. The e a e no Luh-mons e s in any Be gman space Bp(1 ≤p < ∞), so in any Ha dy space Hp(1 ≤p < ∞). P oo . Assume ha ∈Bpand ha Fis a holomo phic unc ion on Dsuch ha F00 = . Then F0is in Hpby Lemma 2.1, so Fis in he disk algeb a, ha is, i can be ex ended con inuously on D: his is asse ed, o ins ance, in [16, Chap e 5, Exe cise 4 9] o p > 1, bu in he case p= 1 a well known esul o P i alo es ablishes ha o h∈H(D) he unc ion h0∈H1i and only i hhas a con inuous ex ension o D ha is absolu ely con inuous on ∂D[16, Theo em 3.11]. I we e a Luh-mons e hen o some sequence (τn)⊂LT(D) wi h τn→1 (n→ ∞) uni o mly on Dwe would ha e F(τn(z)) →g(z) (n→ ∞) in H(D) o he cons an unc ion g∈H(D) wi h g(z) := 1+maxD|F|(z∈D), which is clea ly impossible. Thus, canno be a Luh-mons e , as equi ed. Obse e ha acco ding o he las p oo he an ide i a i es a e o blame o he nonexis ence o Luh-mons e s. Ne e heless, we will be able o deal wi h he exis ence o DN-mons e s in Hpand Bp o any nonnega i e in ege N. In ac , much mo e will be ob ained, see Theo em 2.7. In iew o he nega i e esul p o ided by Theo em 2.2, we now ocus ou a en ion on he sea ch o some sui able condi ion on an ope a o Tde ined on H(G) and on a subspace X⊂H(G) in o de ha T-mons e s can exis in X, i.e., M(T)∩X6=∅. We e en ge ha M(T)∩Xis esidual in Xunde sui able condi ions. This will be made in Lemma 2.3. A e wa ds, wi h he help o a s ong heo em due o Bou don and Shapi o, his lemma is applied in he p oo o Theo em 2.5, in which he exis ence o many holomo phic mons e s ( his is he case T=I) in Ha dy and Be gman spaces is ob ained. We also show how ha ing holomo phic mons e s plus a (pu ely se - heo e ic) so condi ion on a gene al ope a o Tis su icien o ha e T-mons e s in X, see Theo em 2.6. Be o e es ablishing all hese esul s, ecall ha i G⊂Cis a domain and ϕ∈H(D) sa i ies ϕ(D)⊂G hen he composi ion mapping Cϕ: ∈ H(G)7→ ◦ϕ∈H(D) is well de ined and con inuous. In pa icula , ϕcan be any membe o LT(G). Some imes, in he case G=D, he composi ion ope a o Cϕmaps con inuously an F-space (= a comple e linea me ic space) X⊂H(D) in o i sel o e e y holomo phic sel mapping ϕon D. Fo ins ance, his holds o each Ha dy space Hpand each Be gman space Bp(p > 0) due o Li lewood’s subo dina ion heo em, see [26, Chap e 10]. See also [15] o a collec ion o such spaces X. The ollowing auxilia s a emen gi es us a posi i e answe o he p oblem o ex- is ence o mons e s on subspaces in e ms o he exis ence o some kind o hype cyclic sequences. Lemma 2.3. Assume ha Xis an F-space wi h X⊂H(G)and ha Tis an ope a o on H(G)sa is ying he ollowing wo condi ions: 5 (a) Con e gence in Ximplies locally uni o m con e gence. (b) Fo e e y bounda y poin ∈∂G he e is a sequence (τn)⊂LT(G) ending o uni o mly on Dsuch ha he sequence o mappings CτnT:X→H(D) (n∈N) is densely hype cyclic. Then he se { ∈X: is a T-mons e }is esidual in X. P oo . Fix wo se s U(T, g, ε, , V ) and BX(h, α), whe e g∈H(D), h∈X,ε > 0, α > 0, 0 < < 1, V∈O(∂G) and BX(h, α) := { ∈X:d( , h)< α}is an open ball o a ansla ion-in a ian dis ance dcompa ible wi h he opology o X. No e ha each se U(T, g, ε, , V ) is open in H(G) due o he con inui y o T, hence U(T, g, ε, , V )∩Xis open in Xby condi ion (a). On he o he hand, he e a e coun able many se s Un(n∈N) o he ype U(T, g, ε, , V ) such ha M(T) = Tn∈NUn, see [5]. Then M(T)∩X=Tn∈NUn∩X, so M(T)∩Xis a Gδ-subse o X. Since Xis a Bai e space, i su ices o show ha e e y in e sec ion U(T, g, ε, , V )∩Xis dense in Xo , equi alen ly, ha U(T, g, ε, , V )∩BX(h, α)6=∅.(1) Choose any poin ∈V∩∂G and conside he sequence (τn)⊂LT(G) gi en by hypo hesis (b). By dense hype cyclici y, he e is an ∈Xwi h d( , h)< α and a sequence n1< n2<· · · < nk<· · · in Nsuch ha (T )◦τnk→g(k→ ∞) uni o mly on D. Since τn(z)→ (n→ ∞) uni o mly on D, he e is k0∈Nsuch ha τnk0(D)⊂ V∩Gand k(T )◦τnk0−gk D< ε. Hence ∈U(T, g, ε, , V )∩BX(h, α) and (1) is ul illed. Fo ins ance, in he case X=H(G) condi ion (a) is i ially sa is ied and, o T=I, (b) is e en ul illed o e e y ∈∂G by any sequence (τn)⊂LT(G) ending o uni o mly on D, see [7]. 6 The nex asse ion is a e sion o sequences o he Hype cyclici y Compa ison P inciple, see [25, p. 111]. I s p oo is i ial, so i is d opped. The lemma will be used in he p oo o he second pa o Theo em 2.6. Lemma 2.4. Suppose ha X1, X2, X3a e opological spaces in such a way ha X3⊂ X1,X3is dense in X1and he opology o X3is s onge han ha o X1. Assume also ha Sn:X1→X2(n∈N)is a sequence o con inuous mappings wi h he p ope y ha he sequence Sn|X3:X3→X2(n∈N)is densely hype cyclic. Then (Sn)is densely hype cyclic. Theo em 2.5. Assume ha p∈[1,+∞). We ha e: (1) The se { ∈Hp: is a holomo phic mons e }is esidual in Hp. (2) The se { ∈Bp: is a holomo phic mons e }is esidual in Bp. P oo . (1) Condi ions (a)–(b) in Lemma 2.3 should be checked o G=D,X=Hp, T=I. P ope y (a) ollows om he well known es ima e | (z)| ≤ 21/pk kp(1 − |z|)−1/p (z∈D), which holds e en o 0 <p<∞, see o ins ance [16, Chap e 3]. P ope y (b) is mo e delica e. In o de o check i , ix a poin ∈∂Dand conside he unc ion ϕ(z) = z+ 2. T i ially, ϕ∈LT(D) and ϕis no an au omo phism o D. Mo eo e , i s ixed poin s a e (∈∂D) and ∞(6∈ D), he e o e ϕis a non-pa abolic non-au omo phism wi hou ixed poin s in D. Hence, he Linea F ac ional Hype cyclici y Theo em due o Bou - don and Shapi o (see [25, Chap e 7] and [13]; he esul is ob ained o p= 2 bu he p oo equally wo ks o 1 ≤p < ∞because i is ul ima ely based on he ac ha o e e y α∈∂D he collec ion o polynomials anishing a αis dense in Hp, which 7 in u n is a consequence o Beu ling’s app oxima ion heo em, see [16, pp. 113–114]) ells us ha he ope a o Cϕ:Hp→Hpis hype cyclic, so (Cn ϕ) is densely hype cyclic (see Sec ion 1). Bu Cn ϕ=Cτn, whe e τn:= ϕ◦ · · · ◦ ϕ(n- old), i. e., τn(z) = z+ (2n−1) 2n(n∈N). Finally, obse e ha τn(z)→ (n→ ∞) uni o mly on Dand ha Cτn:Hp→H(D) (n∈N) is also densely hype cyclic, because Hpis dense in H(D) and i s no m- opology is s onge han he compac -open one. (2) Choose again G=D,T=Iin Lemma 2.3, wi h X=Bp his ime. P ope y (a) is de i ed om he inequali y (1 − |z|)2| (z)| ≤ || ||p(z∈D, p ≥1), see [26, p. 48]. As o p ope y (b), i is enough o conside he ac ha Cτn: Hp→H(D) (n∈D) is densely hype cyclic (whe e Cτnis as in he p oo o he i s pa ) oge he wi h Lemma 2.4 as applied on X1=Bp,X2=H(D), X3=Hp, Sn=Cτn:Bp→H(D) (n∈N). No e ha Hpis dense in Bpbecause he polynomials a e dense in Bpand Hpcon ains each polynomial. This inishes he p oo . An inmedia e consequence o Theo em 2.5 is ha he se { ∈Hp: is a Cϕ- mons e }is esidual in Hp o e e y au omo phism ϕo D. Indeed, he ope a o T:= Cϕ|Hpmaps homeomo phically Hpon o i sel due o Li lewood’s subo dina ion heo em. Now, he la e se is M(Cϕ)∩Hp=C−1 ϕ(M(I)) ∩Hp=T−1(M(I)∩Hp), which is esidual in Hpbecause M(I)∩Hpis. O cou se, he same holds i Hpis eplaced o Bp. Fo u u e e e ences, we poin ou ha he p oo o he Linea F ac ional Hype - cyclici y Theo em [25] also wo ks o any subsequence (Cnk ϕ) (n1< n2< n3<· · ·) o (Cn ϕ). Theo em 2.6. Assume ha Xis an F-space wi h X⊂H(G)such ha he e is some holomo phic mons e in X. Suppose ha Tis an ope a o on H(G)sa is ying T(X)⊃X. Then he e is some T-mons e in X. 8