Monsters in Hardy and Bergman spaces

Monsters in Hardy and Bergman spaces L. BERNAL–GONZ´ ALEZ and M.C. CALDER´ ON–MORENO Departamento de An´alisis Matem´atico, Facultad de Matem´aticas, Apdo. 1160, Avda. Reina Mercedes, 41080 Sevilla, Spain. E–mails: lb[email protected], [email protected] Abstract A monster in the sense of Luh is a holomorphic function on a simply connected domain in the complex plane such that it and all its derivatives and antiderivatives exhibit an extremely wild behaviour near the boundary. In this paper the Hardy spaces Hpand the Bergman spaces Bp(1 ≤p < ∞) on the unit disk are considered, and it is shown that there are no Luh-monsters in them. Nevertheless, it is proved that T-monsters (as introduced by the authors in an earlier work) can be found in each of these spaces for any finite order linear differential operator T. Key words and phrases: Luh-monster, T-monster, Hardy space, Bergman space, strongly omnipresent operator, differential operator, hypercyclic function. 2000 Mathematics Subject Classification: Primary 30E10. Secondary 30D40, 30H05, 47A16, 47B38. 1 Introduction As soon as the existence of a mathematical entity is established, a natural problem arises: Do such entities exist with additional (even “more perfect”) properties? This 0 has been the line of research which has motivated this paper, this time in the setting of holomorphic functions with “wild” behaviour in the boundary of the open unit disk D={|z|<1}. Suppose that Gis a domain in the complex plane C; by H(G) we denote as usual the Fr´echet space of all holomorphic functions on G, endowed with the compact-open topology. If Gis simply connected then it has been proved in 1985 by W. Luh [21] the existence of a dense set of functions –which he called “monsters”– in H(G) such that all its derivatives and antiderivatives exhibit an extremely wild behaviour near the boundary of G. Such a chaotic property can be expressed in terms of certain generalized cluster sets, introduced by Luh himself. In 1987 Grosse-Erdmann [18], see also [19, Section 4.b], showed that, in fact, there is a residual set of monsters. Further interesting results on this topic can be seen in [22–24]. With the aim of finding operators which are different from those of differentiation and antidifferentiation under whose action there are holomorphic functions with boundary wild behaviour, the authors have recently introduced [5] the notions of T-monsters and strongly omnipresent operators, see Definition 1.1 below. Let us fix some terminology and notation. By ∂G we denote the boundary of a domain G⊂Cin the extended complex plane C∞=C∪ {∞}.Nis the set of positive integers, N0=N∪ {0}and Ris the real line. An operator always refers to a continuous (not necessarily linear) selfmapping. We denote by O(∂G) the set of all open subsets of C∞meeting the boundary of G. If A⊂Cthen Arepresents the closure of A,kfkA:= supz∈A|f(z)|, where fis a complex function defined in A, and LT(A) is the set of all affine linear transformations τ,τ(z) = az +b, such that τ(D)⊂A. In the following definition, we are allowing the point of infinity to be a boundary point of Gwhen Gis unbounded (as in [7, 9, 10]). Observe also that the domain Gis allowed to be non-simply connected in (a)–(c). Definition 1.1. (a) A function f∈H(G) is a holomorphic monster whenever the following universality property is satisfied: For each g∈H(D) and each t∈∂G there exists a sequence (τn) of affine linear transformations with τn(z)→t(n→ ∞) uniformly on Dand τn(D)⊂G(n∈N) such that f(τn(z)) →g(z) (n→ ∞) locally uniformly in D. 1 (b) Let T:H(G)→H(G) be an operator. Then a function f∈H(G) is a T– monster if Tf is a holomorphic monster. The set of T–monsters is denoted by M(T). (c) An operator T:H(G)→H(G) is strongly omnipresent if for all g∈H(D), ε > 0, r∈(0,1) and V∈O(∂G) the set U(T, g, ε, r, V ) := {f∈H(G) : there exists some τ∈LT(V∩G) such that k(Tf)◦τ−gkrD< ε} is dense in H(G). (d) If Gis simply connected, then a function f∈H(G) is a Luh–monster whenever every derivative f(n)(n∈N0) and every antiderivative f(−n)(n∈N) is a holomorphic monster. Observe that fis a holomorphic monster if and only if it is an I-monster (I:= the identity operator), and that fis a Luh-monster if and only if fis simultaneously aDN-monster (for all N∈N0) and a D−N a-monster (for all N∈Nand all a∈G). Here DN(N∈N0) is the differentiation operator DNf=f(N),D0=I, and D−N a is the antidifferentiation operator given by D−N af:= the unique antiderivative Fof order Nof fsuch that F(a) = · · · =F(N−1)(a) = 0. It happens that an operator Ton H(G) is strongly omnipresent if and only if the set M(T) is residual (see [5, Theorem 2.2]). Hence Grosse-Erdmann [18, Kapitel 3] had showed in fact that every DNand every D−N ais strongly omnipresent. He and the authors have identified several kinds of strongly omnipresent operators, including infinite order differential and antidifferential operators, integral operators, composition and multiplication operators [5, 9, 10]. For other kinds of operators –also introduced by the authors– under whose action certain functions have some type of boundary chaotic behaviour, the reader is referred to [1] (omnipresent operators), [2, 6, 14] (DI-operators) and [7] (totally omnipresent operators), see also [8]. It happens that every totally omnipresent operator is strongly omnipresent and DI, and that if an operator is either strongly omnipresent or DI then it is omnipresent. By using totally omnipresent operators, the authors have recently 2 proved (see [7]) that there is a dense linear manifold in H(G) all of whose nonzero functions are Luh-monsters. We will need some terminology about universality (see [19] for an excellent survey, updated till 1999). If Xand Yare (Hausdorff) topological vector spaces over the same field K(= Ror C) and Tn:X→Y(n∈N) is a sequence of continuous linear mappings, then (Tn) is said to be hypercyclic (or universal) if and only if there is a vector x∈X, called also hypercyclic for (Tn), such that the orbit {Tnx:n∈N} is dense in Y. The sequence (Tn) is called densely hypercyclic whenever the set HC((Tn)) of hypercyclic vectors for (Tn) is dense. If X=Yand Tis a linear operator on Xthen Tis called hypercyclic if and only if the sequence (Tn) of iterates is hypercyclic. It is easy to see that in such a case (Tn) is indeed densely hypercyclic. The existence of T-invariant dense linear manifolds of hypercyclic vectors for each hypercyclic linear operator Ton a (real or complex) locally convex space was shown by Herrero, Bourdon and B`es [11, 12, 20] (see also [4] for the additional property of maximal algebraical cardinality to such manifolds). In 1999 the first author extended this result to hypercyclic sequences of mappings, see [3] and Theorem 2.6. For the sake of convenience, we will keep in this work the notion of hypercyclicity even when the spaces X, Y and the mappings T, Tnare not linear. The aim of this paper is to study the existence of Luh-monsters and in general of T-monsters in (not necessarily closed) subspaces of H(G), mainly in the maybe most emblematic spaces of analytic functions on G=D, namely, the Hardy spaces Hpand the Bergman spaces Bp(1 ≤p < ∞). Recall that Hpis the class of all functions f∈H(D) satisfying kfkp:= sup 0<r<1Z2π 0 |f(reiθ)|pdθ 2π1/p <∞, while Bpis the class of all f∈H(D) satisfying kfkp:= ZD |f(z)|pdA(z)1/p <∞. Each one becomes a Banach space under the corresponding norm k · kp. Recall also that Hp⊂Bpwith continuous inclusion. We have denoted by dA(z) the area measure on Dnormalized so that the area of Dis 1. The existence of dense linear manifolds of holomorphic monsters in these spaces is also considered. 3 2 Monsters in subspaces of H(D) Up to date, all results about monsters have been addressed to state their existence, with success except for very special examples. However, in the current setting, we are on the point to obtain a general statement of non-existence of classical monsters in function spaces, see Theorem 2.2 below. Before this, we establish an auxiliary lemma whose content is probably well known. Since we have not been able to find a reference for it, we provide with an elementary proof. Lemma 2.1. If f∈H(D)and f0∈Bpfor some p∈[1,∞)then f∈Hp. Proof. By hypothesis, RD|f0(z)|pdA(z)<∞. Passing to polar coordinates, we get R1 0R2π 0|f0(seiθ)|ps dsdθ < ∞. Since |f0|pis Lebesgue-integrable on a neighbourhood of the origin, we can drop the factor s, that is, Z1 0Z2π 0 |f0(seiθ)|pdsdθ < ∞. It is evident that we can suppose f(0) = 0. Then f(reiθ) = Rr 0f0(seiθ)eiθ ds for all r∈[0,1) and all θ∈[0,2π]. Hence, by the H¨older inequality, |f(reiθ)|p≤Zr 0 |f0(seiθ)|dsp ≤rp−1Zr 0 |f0(seiθ)|pds ≤Z1 0 |f0(seiθ)|pds. Discarding the terms in the middle, an integration over [0,2π] yields the desired result. Theorem 2.2. There are no Luh-monsters in any Bergman space Bp(1 ≤p < ∞), so in any Hardy space Hp(1 ≤p < ∞). Proof. Assume that f∈Bpand that Fis a holomorphic function on Dsuch that F00 =f. Then F0is in Hpby Lemma 2.1, so Fis in the disk algebra, that is, it can be extended continuously on D: this is asserted, for instance, in [16, Chapter 5, Exercise 4 9] for p > 1, but in the case p= 1 a well known result of Privalov establishes that for h∈H(D) the function h0∈H1if and only if hhas a continuous extension to Dthat is absolutely continuous on ∂D[16, Theorem 3.11]. If fwere a Luh-monster then for some sequence (τn)⊂LT(D) with τn→1 (n→ ∞) uniformly on Dwe would have F(τn(z)) →g(z) (n→ ∞) in H(D) for the constant function g∈H(D) with g(z) := 1+maxD|F|(z∈D), which is clearly impossible. Thus, fcannot be a Luh-monster, as required. Observe that according to the last proof the antiderivatives are to blame for the nonexistence of Luh-monsters. Nevertheless, we will be able to deal with the existence of DN-monsters in Hpand Bpfor any nonnegative integer N. In fact, much more will be obtained, see Theorem 2.7. In view of the negative result provided by Theorem 2.2, we now focus our attention on the search of some suitable condition on an operator Tdefined on H(G) and on a subspace X⊂H(G) in order that T-monsters can exist in X, i.e., M(T)∩X6=∅. We even get that M(T)∩Xis residual in Xunder suitable conditions. This will be made in Lemma 2.3. Afterwards, with the help of a strong theorem due to Bourdon and Shapiro, this lemma is applied in the proof of Theorem 2.5, in which the existence of many holomorphic monsters (this is the case T=I) in Hardy and Bergman spaces is obtained. We also show how having holomorphic monsters plus a (purely settheoretic) soft condition on a general operator Tis sufficient to have T-monsters in X, see Theorem 2.6. Before establishing all these results, recall that if G⊂Cis a domain and ϕ∈H(D) satifies ϕ(D)⊂Gthen the composition mapping Cϕ:f∈ H(G)7→ f◦ϕ∈H(D) is well defined and continuous. In particular, ϕcan be any member of LT(G). Sometimes, in the case G=D, the composition operator Cϕmaps continuously an F-space (= a complete linear metric space) X⊂H(D) into itself for every holomorphic selfmapping ϕon D. For instance, this holds for each Hardy space Hpand each Bergman space Bp(p > 0) due to Littlewood’s subordination theorem, see [26, Chapter 10]. See also [15] for a collection of such spaces X. The following auxiliar statement gives us a positive answer to the problem of existence of monsters on subspaces in terms of the existence of some kind of hypercyclic sequences. Lemma 2.3. Assume that Xis an F-space with X⊂H(G)and that Tis an operator on H(G)satisfying the following two conditions: 5 (a) Convergence in Ximplies locally uniform convergence. (b) For every boundary point t∈∂G there is a sequence (τn)⊂LT(G)tending to t uniformly on Dsuch that the sequence of mappings CτnT:X→H(D) (n∈N) is densely hypercyclic. Then the set {f∈X:fis a T-monster}is residual in X. Proof. Fix two sets U(T, g, ε, r, V ) and BX(h, α), where g∈H(D), h∈X,ε > 0, α > 0, 0 < r < 1, V∈O(∂G) and BX(h, α) := {f∈X:d(f, h)< α}is an open ball for a translation-invariant distance dcompatible with the topology of X. Note that each set U(T, g, ε, r, V ) is open in H(G) due to the continuity of T, hence U(T, g, ε, r, V )∩Xis open in Xby condition (a). On the other hand, there are countable many sets Un(n∈N) of the type U(T, g, ε, r, V ) such that M(T) = Tn∈NUn, see [5]. Then M(T)∩X=Tn∈NUn∩X, so M(T)∩Xis a Gδ-subset of X. Since Xis a Baire space, it suffices to show that every intersection U(T, g, ε, r, V )∩Xis dense in Xor, equivalently, that U(T, g, ε, r, V )∩BX(h, α)6=∅.(1) Choose any point t∈V∩∂G and consider the sequence (τn)⊂LT(G) given by hypothesis (b). By dense hypercyclicity, there is an f∈Xwith d(f, h)< α and a sequence n1< n2<· · · < nk<· · · in Nsuch that (Tf)◦τnk→g(k→ ∞) uniformly on rD. Since τn(z)→t(n→ ∞) uniformly on D, there is k0∈Nsuch that τnk0(D)⊂ V∩Gand k(Tf)◦τnk0−gkrD< ε. Hence f∈U(T, g, ε, r, V )∩BX(h, α) and (1) is fulfilled. For instance, in the case X=H(G) condition (a) is trivially satisfied and, for T=I, (b) is even fulfilled for every t∈∂G by any sequence (τn)⊂LT(G) tending to tuniformly on D, see [7]. 6 The next assertion is a version for sequences of the Hypercyclicity Comparison Principle, see [25, p. 111]. Its proof is trivial, so it is dropped. The lemma will be used in the proof of the second part of Theorem 2.6. Lemma 2.4. Suppose that X1, X2, X3are topological spaces in such a way that X3⊂ X1,X3is dense in X1and the topology of X3is stronger than that of X1. Assume also that Sn:X1→X2(n∈N)is a sequence of continuous mappings with the property that the sequence Sn|X3:X3→X2(n∈N)is densely hypercyclic. Then (Sn)is densely hypercyclic. Theorem 2.5. Assume that p∈[1,+∞). We have: (1) The set {f∈Hp:fis a holomorphic monster}is residual in Hp. (2) The set {f∈Bp:fis a holomorphic monster}is residual in Bp. Proof. (1) Conditions (a)–(b) in Lemma 2.3 should be checked for G=D,X=Hp, T=I. Property (a) follows from the well known estimate |f(z)| ≤ 21/pkfkp(1 − |z|)−1/p (z∈D), which holds even for 0 <p<∞, see for instance [16, Chapter 3]. Property (b) is more delicate. In order to check it, fix a point t∈∂Dand consider the function ϕ(z) = z+t 2. Trivially, ϕ∈LT(D) and ϕis not an automorphism of D. Moreover, its fixed points are t(∈∂D) and ∞(6∈ D), therefore ϕis a non-parabolic non-automorphism without fixed points in D. Hence, the Linear Fractional Hypercyclicity Theorem due to Bourdon and Shapiro (see [25, Chapter 7] and [13]; the result is obtained for p= 2 but the proof equally works for 1 ≤p < ∞because it is ultimately based on the fact that for every α∈∂Dthe collection of polynomials vanishing at αis dense in Hp, which 7 in turn is a consequence of Beurling’s approximation theorem, see [16, pp. 113–114]) tells us that the operator Cϕ:Hp→Hpis hypercyclic, so (Cn ϕ) is densely hypercyclic (see Section 1). But Cn ϕ=Cτn, where τn:= ϕ◦ · · · ◦ ϕ(n-fold), i. e., τn(z) = z+ (2n−1)t 2n(n∈N). Finally, observe that τn(z)→t(n→ ∞) uniformly on Dand that Cτn:Hp→H(D) (n∈N) is also densely hypercyclic, because Hpis dense in H(D) and its normtopology is stronger than the compact-open one. (2) Choose again G=D,T=Iin Lemma 2.3, with X=Bpthis time. Property (a) is derived from the inequality (1 − |z|)2|f(z)| ≤ ||f||p(z∈D, p ≥1), see [26, p. 48]. As for property (b), it is enough to consider the fact that Cτn: Hp→H(D) (n∈D) is densely hypercyclic (where Cτnis as in the proof of the first part) together with Lemma 2.4 as applied on X1=Bp,X2=H(D), X3=Hp, Sn=Cτn:Bp→H(D) (n∈N). Note that Hpis dense in Bpbecause the polynomials are dense in Bpand Hpcontains each polynomial. This finishes the proof. An inmediate consequence of Theorem 2.5 is that the set {f∈Hp:fis a Cϕmonster}is residual in Hpfor every automorphism ϕof D. Indeed, the operator T:= Cϕ|Hpmaps homeomorphically Hponto itself due to Littlewood’s subordination theorem. Now, the latter set is M(Cϕ)∩Hp=C−1 ϕ(M(I)) ∩Hp=T−1(M(I)∩Hp), which is residual in Hpbecause M(I)∩Hpis. Of course, the same holds if Hpis replaced to Bp. For future references, we point out that the proof of the Linear Fractional Hypercyclicity Theorem [25] also works for any subsequence (Cnk ϕ) (n1< n2< n3<· · ·) of (Cn ϕ). Theorem 2.6. Assume that Xis an F-space with X⊂H(G)such that there is some holomorphic monster in X. Suppose that Tis an operator on H(G)satisfying T(X)⊃X. Then there is some T-monster in X. 8