Weighted Spaces of Holomorphic Functions and ...

Weighted Spaces of Holomorphic Functions and Operators between them José Bonet Abstract This survey contains part of the material which was presented at the Seminar of Mathematical Analysis of the Faculty of Mathematics of the University of Sevilla in October 2002. We discuss weighted Banach spaces and weighted inductive limits of spaces of holomorphic functions defined on an open subset G of CN and operators defined between them. We report on recent work about the structure of these spaces and operators defined on them.

1

Weighted Banach spaces of holomorphic functions

Let G be an open subset of CN , N ∈ N, and let v : G → R be a continuous and strictly positive weight on G. We define the following weighted Banach spaces of holomorphic functions on G Hv(G) := {f ∈ H(G); ||f ||v := sup v(z)|f (z)| < ∞}, z∈G

Hv0 (G) := {f ∈ H(G); v|f | vanishes at ∞ on G}. We recall that a function g vanishes at infinity on G if for every ε > 0 there is a compact subset K of G such that |g(z)| < ε if z ∈ / K. We assume in what follows that the norm ||δz || of the Dirac measure, δz (f ) := f (z), z ∈ G as an element of Hv(G)0 is strictly positive. If G is an open connected subset of C, this happens if Hv(G) 6= {0}. If G = D and v(z) = 1 for each z ∈ D, then Hv0 (D) = {0}. If G = C and v(z) = |z| for each z ∈ C, then Hv0 (C) consists only of the constants. On the other hand, it is easy to show that if the dimension of Hv0 (G) is larger or equal than 2, then Hv0 (G) separates points of G. This happens for example if G is a bounded domain in C and Hv0 (G) 6= {0}. 2000 Mathematics Subject Classification. Primary 46E10, 46B03 Key words and phrases. Weighted Banach spaces, holomorphic functions, multiplication operators, sampling sets, weighted inductive limits It is a pleasure to thank Luis Bernal and Carmen Calderón and all the research group for their kind invitation to give this series of lectures in Sevilla in October 2002.

1

Banach spaces of the type mentioned above appear naturally in the study of growth conditions of analytic functions and have been considered in many papers. We refer to [4, 5, 8, 35, 53]. Composition operators on weighted Banach spaces of this type when G = D have been studied in [13, 14, 23, 47, 55]. Pointwise multiplication operators were considered in [15, 22], and sampling and interpolation in these spaces in [25]. Banach spaces Hv0 (G) for radial weights v on the disc G = D have been studied widely in the literature. See e.g. [35, 38, 39, 53, 54]. In [41] Lusky has found all the isomorphy classes of the spaces Hv(D) when the radial weight v is of ”moderate decay”. The spaces Hv0 (G) for radial weights on the complex plane G = C were investigated by Galbis [26]. Lusky has adapted his methods for the case of spaces of entire functions in [44]; see also [42]. Weighted inductive limits of spaces of holomorphic functions (and in particular of weighted Banach spaces as defined above) arise in several areas of analysis like convolution equations, theory of ultradistributions, spectral theory and complex analysis. The problem of the projective description of these inductive limits and their topology has been systematically studied in several articles [2, 3, 7]. The projective description allows direct computation and estimates as required in applications. Bierstedt [3] has recently written a survey about countable inductive limits of weighted Banach spaces Hv(G) or Hv0 (G) which includes open problems. Several instances in which weighted inductive limits play an important role in recent investigations are discussed in the last three sections. Our notation for complex analysis, Banach spaces and functional analysis is standard. We refer the reader to [31, 33, 37, 46, 48, 49]. If v is a (continuous and strictly positive) weight on G, its associated weight is defined by v˜(z) := 1/||δz ||Hv(G)0 . By our assumption above, v˜(z) is finite for every z ∈ G. Moreover v ≤ v˜ on G, 1/˜ v is continuous and subharmonic, and the Banach spaces Hv(G) and H v˜(G) coincide isometrically. A weight v is called essential if there is C ≥ 1 such that v ≤ v˜ ≤ Cv on G. The continuity of v˜ can be deduced from the following result which might be of independent interest. It is a consequence of a recent theorem by Grosse-Erdmann [30]. Lemma 1.1 If E is a barrelled locally convex space of holomorphic functions (for example a Banach space) with a topology which is finer than the pointwise convergence, then the map ∆ : G → Eb0 defined by ∆(z) := δz is holomorphic, hence continuous. Examples 1.2 (1) If G = C and 1/v(z) = max(1, |z|n+p ), z ∈ C, n ∈ N, 0 < p < 1, then 1/˜ v (z) = max(1, |z|n ). (2) If G = D or G = C and v is radial, then v˜ is also radial. (3) ([13]) If G = D, v is radial and limr→1− v(r) = 0, then limr→1− v˜(r) = 0. (4) ([14]) A radial and non-increasing weight v on D is essential if and only if it is equivalent to a log-convex radial weight w on D. We recall that the weight w is log-convex if the function t → − log w(et ) is convex. 2

(5) Every normal weight on D in the sense of Shields and Williams is essential. We refer the reader to [5, 54]. (6) Essential weights on a simply connected domain in C which depend polynomially on the distance to the boundary have been studied in [25]. (7) Seip proved in 1998 that if v is a weight on D which is C 2 and −∆ log v(z) behaves like (1 − |z|2 )2 near the boundary, then v is essential; see [25]. Here ∆ denotes the Laplacian. Examples 1.3

(1) Concrete examples of weights on G = D.

(1.1) v(r) = (1 − r)p , 0 < p < ∞. 1 (1.2) v(r) = exp − (1−r) , q > 0. q (2) Concrete examples of weights on G = C. (2.1) v(r) = exp(−rp ), p > 0. (2.2) v(r) = exp(e−r ). An open set G in CN is called balanced if λz ∈ G for each z ∈ G and each |λ| ≤ 1. A weight v on a balanced open set G is called radial if v(λz) = v(z) if |λ| = 1. The following result follow from [4, 8]. Theorem 1.4 Suppose that v is a radial weight on a balanced open subset G of CN . If Hv0 (G) contains the polynomials, then the space Hv0 (G) has the metric approximation property, the polynomials are dense in it, and the space Hv(G) is canonically isometric to the bidual of Hv0 (G). Moreover, the space Hv(G) is an L∞ -space if and only if the space Hv0 (G) is an L∞ -space if and only if Hv0 (G) is isomorphic to the sequence space c0 . Lusky [38, 39] has characterized the spaces of type Hv0 (D) which are isomorphic to c0 in terms of the weight in case that the weight v does not grow too fast. Theorem 1.5 Let v be a radial non-increasing weight on D such that limr→1− v(r) = 0. Suppose that the following condition holds: (∗)

sup n

v(1 − 2−n ) < ∞. v(1 − 2−n−1 )

The space Hv0 (D) is isomorphic to c0 if and only if for some k ∈ N (∗∗)

v(1 − 2−n−k ) lim sup < 1. −n ) n→∞ v(1 − 2 3

A radial and non-increasing weight v on D satisfies the conditions (*) and (**) of Lusky of Theorem 1.5 if and only if it is a normal weight in the sense of Shields and Williams [54, 53]. We refer the reader to [25] for details. The example 1.3 (1.1) satisfies the conditions (*) and (**). The weight v(r) = (1 − log(1 − r))q , q < 0, 0 < r < 1, satisfies condition (*) but not the condition (**). Accordingly the space Hv0 (D) is not isomorphic to c0 . The first example of this type is due to Kaballo [35]. The example 1.3 (1.2) is an “exponential” weight which does not satisfy the condition (*). Nevertheless Lusky proved in [43] that Hv0 (D) is isomorphic to c0 for the weight v(z) = exp(−1/(1 − |z|)), z ∈ D. Lusky [41] has obtained a complete isomorphic classification of the spaces Hv(D) if the weight v is radial non-increasing and satisfies the condition (*). Theorem 1.6 Suppose that the radial non-increasing weight v on D satisfies the condition (*). Then (1) if v satisfies the condition (**), then Hv(D) is isomorphic to l∞ . (2) if v does not satisfies the condition (**), then Hv(D) is isomorphic to H ∞ . In the case of radial non-increasing weights on the complex plane C such that Hv0 (C) contains the polynomials, Lusky [44] proved that if v is one of the weights in 1.3 (2), then Hv0 (C) is isomorphic to c0 . The case of 1.3 (2.1) for p ∈ N was due to Galbis [26]. Galbis [26] constructed also radial weights on C such that Hv0 (C) is not an L∞ -space. It was an open problem of Garling and Wojtaszczyk from 1995 whether the monomials form a Schauder basis for Hv0 (C) for the weight v(z) = exp(−|z|p ), z ∈ C. Lusky proved in [42] that this is not the case. This research was continued in [9]. An important tool in the proofs of the results of Lusky is the use of the operators which appear as convolution with de la Vallée Poussin kernel. We refer the reader to his articles. He has also extended some of his results for weighted Bergman spaces [40]. Not much seemed to be known about the structure of weighted Banach spaces of holomorphic functions defined on arbitrary open subsets G of CN . A method of Kalton and D. Werner has been used by E. Wolf and the author to show the following result. Theorem 1.7 Let G be an open subset of CN , N ≥ 1, and let v be a strictly positive and continuous weight on G. Then the space Hv0 (G) is isomorphic to a closed subspace of c0 . Corollary 1.8 Let G be an open subset of CN , N ≥ 1, and let v be a strictly positive and continuous weight on G. If the space Hv0 (G) is infinite dimensional, then Hv0 (G) and Hv(G) are not reflexive. The corollary 1.8 follows from [14, Theorem 1] if G is an open connected domain in C such that its complement in the Riemann sphere has no one-point component.

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2

Composition Operators

Composition operators on various spaces of analytic functions on the unit disc have been studied very thoroughly by a number of authors; cf. the books of Cowen, MacCluer [24] and of Shapiro [52]. Composition operators constitute a very active area of research as a search in the databases of Mathematical Reviews or Zentralblatt shows. In this section we only report on joint work with Doma´ nski, Lindström and Taskinen about weighted composition operators defined on Banach spaces Hv(D) and Hv0 (D) as defined in the previous section and directly related work; see [13, 14, 16, 17, 23, 47, 55]. Through the section we suppose that all the weights v are radial, non-increasing on D and satisfy that limr→1− v(r) = 0. We denote by ϕ : D → D an analytic map. The composition operator Cϕ : H(D) → H(D) is defined by Cϕ (f ) := f ◦ ϕ. Proposition 2.1 The following conditions are equivalent for the composition operator Cϕ : Hv(D) → Hw(D): (1) The operator Cϕ is continuous. w(z) v˜(ϕ(z))

< ∞.

||ϕ(z)n ||w ||z n ||v

< ∞.

(2) supz∈D (3) supn

Proposition 2.2 The following conditions are equivalent for the composition operator Cϕ : Hv(D) → Hw(D): (1) The operator Cϕ is (weakly) compact. (2) lim|z|→1

w(z) v˜(ϕ(z))

(3) limn→∞

||ϕ(z)n ||w ||z n ||v

= 0. = 0.

Estimates of the essential norm of a composition operator Cϕ : Hv(D) → Hw(D) can be seen in [14]. Related results and consequences for composition operators on Bloch spaces are given in [23, 47]. Examples of non-nuclear compact composition operators on Hv(D) are due to Taskinen [55]. Proposition 2.3 Let ϕ be given. The following holds: (1) Cϕ : Hv(D) → Hv(D) is continuous for all radial non-increasing weights v if and only if there is 0 < s < 1 such that |ϕ(z)| ≤ |z| for all |z| ≥ s. (2) Cϕ : Hv(D) → Hv(D) is compact for all radial non-increasing weights v if and only if ϕ(D) ⊂ sD for some 0 < s < 1.

5

Proposition 2.4 A radial non-increasing weight v satisfies that the operator Cϕ : Hv(D) → Hv(D) is continuous for every ϕ if and only if the weight v˜ satisfies the condition (*). If ϕ(z) = (z + 1)/2, z ∈ D, and v(z) = exp(−1/(1 − |z|)), z ∈ D, the composition operator Cϕ is not continuous on Hv(D). The case of composition operators on weighted spaces of vector valued holomorphic functions has been discussed in [16, 17] where previous work by Liu, Saksman and Tylli is continued.

3

Multiplication Operators

Let ϕ ∈ H(D), ϕ 6= 0. The linear operator Mϕ : H(D) −→ H(D) f 7→ ϕf is called a pointwise multiplication operator. Pointwise multiplication operators between different spaces of analytic functions have been studied by many authors, like Axler, Luecking, McDonald and Sundberg and Vukotic. We refer the reader to [15, 22, 57] for the precise references. In this section we present some results about pointwise multiplication operators Mϕ defined on weighted Banach spaces of holomorphic functions Hv(D). The results are taken from [15, 22]. (1) Mϕ (Hv(D)) ⊂ Hv(D) if and only if Mϕ is continuous if and only if ϕ ∈ H ∞ , and in this case kϕk∞ = kMϕ k (2) Mϕ is injective. (3) Mϕ is an isomorphism on Hv(D) if and only if 1/ϕ ∈ H ∞ . (4) Mϕ is a Fredholm operator if and only if there exists ε > 0 such that |ϕ(z)| ≥ ε whenever 1 − ε ≤ |z| < 1. Recall that an operator T is called Fredholm if ImT is closed and finite codimensional and KerT is finite dimensional. The most interesting problem in this context is to establish when is Mϕ an isomorphism into or, equivalently, when it has closed range. (5) The operator Mϕ : H ∞ −→ H ∞ (which is the case v ≡ 1) has closed range if and only if the radial extension ϕ∗ is bounded away from 0 at ∂D, i.e. if there exists ε > 0 such that |ϕ∗ | ≥ ε almost everywhere on ∂D. (6) If ϕ is an outer function or ϕ ∈ A(D) (the disc algebra), then Mϕ : Hv(D) → Hv(D) is an isomorphism into if and only if Mϕ is a Fredholm operator.

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Theorem 3.1 ([15]) Let v be a radial weight which verifies the conditions (*) and (**) of Lusky. The operator Mϕ : Hv(D) → Hv(D) has a closed range if and only if ϕ = hb, where h is invertible and b is a finite product of Blaschke products whose sequence of zeros is interpolating in H ∞ (see Section 4). 1 (7) Let v(z) = exp(− 1−|z| ). Then Mϕ is an isomorphism into if and only if Mϕ is a Fredholm operator

Cicho´ n and Seip [22] improve previous results of Bogalska [10] which extended (7) to show the following result which should be compared with Theorem 3.1. Theorem 3.2 ([22]) . Let v ∈ C 2 (D) be a radial weight such that −(1 − |z|2 )2 4 log v(z) −→ +∞ if |z| −→ 1− . The operator Mϕ : Hv(D) → Hv(D) is an isomorphism into if and only if ϕ = hb, where h is invertible and b is a finite Blaschke product.

4

Interpolation and Sampling

In this section we explain recent work by Doma´ nski, Lindström [25]. They use ideas and results of Seip [50, 51]. The work of Seip can be seen in [31]. It was continued later by Berndtsson, Ortega-Cerdá, Schuster and others. The precise references can be found in [25]. Let v be a continuous and strictly positive weight on D. For a given sequence Γ = (zn )n ⊂ D, we define T : Hv(D) −→ l∞ . f 7→ (f (zn )v(zn ))n The sequence Γ is called a set of interpolation for v if T is surjective, a set of linear interpolation for v if T has continuous and linear right inverse and a sampling set for v if T is a monomorphism. Every sampling set is a set of uniqueness for Hv(D). Compare with the Weierstrass theorem. Seip [50] characterized the sets of interpolation and sampling for A−p := Hvp (D) if vp (z) = (1 − |z|2 )p , p > 0, in terms of certain densities. The classical interpolation problem in H ∞ (Hv(D) for v ≡ 1) was solved by Carleson in 1952: Γ = (zn )n is a set of interpolation for H ∞ if and only if there exists δ > 0 such that for each k ∈ N Y zn − zk 1 − z n zk ≥ δ. n6=k The pseudohyperbolic metric in D is defined by ω−z , z, ω ∈ D. ρ(z, ω) := 1 − ωz 7

A sequence Γ = (zn )n is said to be (ρ-) uniformly discrete if inf n6=k ρ(zn , zk ) > 0. For 1/2 < r < 1 and ϕω (z) := (ω − z)/(1 − ωz) we define D(Γ, r) :=

log

1 1−r

−1

X 1 Uv , then Γ is a set of sampling for v. (d) If Lv > 0 and Γ is a set of sampling for v, then there exists Γ0 ⊂ Γ uniformly discrete such that D− (Γ0 ) > Lv . Corollary 4.2 1. If v is essential and Uv = 0, then every set of interpolation Γ for v is uniformly discrete and D+ (Γ) = 0. An example is given by the weight ε e v(z) = log , ε < 0. 1 − |z| 2. If v is essential and 0 < Lv = Uv < ∞ (which is equivalent to v being a normal weight), then 8

(a) Γ is a set of interpolation for v if and only if Γ is uniformly discrete and D+ (Γ) < Uv . (b) Γ is a a set of sampling for v if and only if there exists Γ0 ⊂ Γ uniformly discrete such that D− (Γ) > Lv . In particular, if v(z) = (1 − |z|2 )p log(e/(1 − |z|))ε , then Hv(D) and Hvp (D) have the same sets of interpolation and sampling, although they do not coincide as Banach spaces.

5

Weighted inductive limits of spaces of holomorphic functions.

Given a decreasing sequence V = (vn )n of weights on G we have Hvn (G) ⊂ Hvn+1 (G) and H(vn )0 (G) ⊂ H(vn+1 )0 (G) with continuous inclusion. We may form the inductive limits V H(G) := indHvn (G) and V0 H(G) := indH(vn )0 (G). We recall that the inductive limit E = indn En is the union E of the spaces En endowed with the finest locally convex topology such that all the inclusions En ⊂ E are continuous. The most important and motivating problem about V H(G) and V0 H(G) concerns the description of their topology. In order to describe the topology of these spaces Bierstedt, Meise and Summers introduced the following system of weights associated with V V := {v : G → [0, ∞[: v upper semicontinuous and v/vn bounded in G for each n ∈ N}. The projective hulls of the weighted inductive limits are defined as HV (G) := {f ∈ H(G) : pv f := sup v(z)|f (z)| < ∞ for all v ∈ V } z∈G

HV 0 (G) := {f ∈ H(G) : v(z)|f (z)| vanishes at ∞ on G for all v ∈ V } and they are endowed with their canonical locally convex topology. The space HV 0 (G) is a closed subspace of HV (G), and V H(G) ,→ HV (G) and V H0 (G) ,→ HV 0 (G) with continuous inclusions. Bierstedt, Meise and Summers [7] showed that V H(G) = HV (G) algebraically and that V H(G) is complete. Moreover they proved the following very useful criterion. Theorem 5.1 (Bierstedt, Meise and Summers [7]) If V satisfies the condition (S) that for all n there exists m > n such that vm /vn vanishes at ∞ on G, then V H(G) = V0 H(G) = HV (G) = HV 0 (G) hold algebraically and topologically. Taskinen, in a stay in Paderborn in 1993, had an idea to construct an example which showed that V H(G) need not coincide with HV (G) topologically. The counterexample was published later in [20]. Since 1995 the author has participated in the construction of other (simpler) counterexemples. The references can be seen in [3]. 9

Examples 5.2 (1) Let N ≥ 1 and let B be the euclidean unit ball of CN . For n ∈ N we consider the system of weights vn (z) = (1 − |z|)n . Clearly the sequence V = (vn )n is decreasing on B. The space V H(B) = indHvn (B) is the smallest algebra of holomorphic functions on B which contains H ∞ (B) and is closed under differentiation. For N = 1 it was studied by Korenblum in 1975. It is denoted by A−∞ . i.e. we can write A−∞ = indp>0 A−p , A−p := {f ∈ H(D) : kf kp := sup(1 − |z|2 )p |f (z)| < ∞} Observation Consider the weights vn (z) = (1 − |z|)n and wn (z) = (1 − |z|2 )n , and the systems of weights V = (vn )n and W = (wn )n . The spaces V H(B) and W H(B) are canonically isomorphic since the inequality 1 − r < 1 − r2 < 2(1 − r) holds for each 0 < r < 1. (2) Let p : C −→ [0, ∞[ be a subharmonic radial function such that log(1+|z|2 ) = o(p(z)), p(2z) = O(p(z)) as |z| tends to ∞. For the weights vn (z) = exp(−np(z)) for z ∈ C, the radial Hörmander algebra on C is defined by V H(C) = Ap (C) := indn Hvn (C). The polynomials are dense in Ap (C) and Ap (C) is closed under differentiation. As a distinguished example we consider p(z) = |z|, and we set in this case Exp(C) := Ap (C). Its elements are the functions of exponential type. This space is isomorphic to the strong dual of the space of entire functions, i.e. Exp(C) = H(C)0b . These spaces and its relevance for interpolation are discussed in Chapter 2 of [1]. There are still several open problems concerning the projective description of weighted inductive limits of spaces of holomorphic functions. We refer the interested reader to [3]. Recent progress has been obtained in the context of weighted (LF)-spaces of holomorphic functions in [6, 19]. Instead of discussing these developments, we present in the next three sections different instances in which the weighted inductive limits of spaces of holomorphic functions play a relevant role: the hypercyclicity of the differentiation operator, the continuity of the Bergman projection and the weakly sufficient sets in the Korenblum algebra A−∞ of holomorphic functions on the disc.

6

Hyperciclicity of the differentiation operator

Definition 6.1 (1) Let E be a locally convex space. An operator T ∈ L(E) is called hypercyclic if there exists x ∈ E such that its orbit Orb(T, x) := {x, T (x), T 2 (x), · · · } is dense in E. (2) An operator T ∈ L(E) is called chaotic if it is transitive (for all U, V open subsets of E there exists n ∈ N such that T n (U ) ∩ V 6= ∅) and the set of periodic points of T is dense in E. 10

Hypercyclic and chaotic operators on Banach or Fréchet spaces have been recently thoroughly investigated by Ansari, Bernal, Bès, Bourdon, Godefroy, Grosse-Erdmann, Montes, Peris, Shapiro and many others. We refer the reader to the survey articles [29, 18]. In this section we present some results from [11] about partial differential operators defined on weighted inductive limits of holomorphic functions closed under differentiation mentioned in the Examples 5.2. These spaces are neither metrizable nor Baire. Theorem 6.2 ([11]) If P (D) is a partial differential operator with constant coefficients which is not a scalar multiple of the identity, then P (D) is hypercyclic and chaotic on A−∞ (B). Theorem 6.3 ([11]) (1) The operator of differentiation D is not hypercyclic on Ap (C) if p(r) = o(r) as r tends to ∞. (2) The operator of differentiation D is hypercyclic and chaotic on Ap (C) if r = O(p(r)) as r tends to ∞. Idea of the proof. (1) If D is hypercyclic on Ap (C) then D is hypercyclic on H(C) with an hypercyclic vector f ∈ Ap (C). This contradicts a theorem of Grosse-Erdmann [28] which asserts that there is no hypercyclic vector f for the operator of differentiation on H(C) such that |f (z)| = O(|z|−1/2 exp |z|) as z goes to ∞. (2)Apply the hypercyclicity criterion [27, 1.5] for D on the Banach space Hv0 (C) := {f ∈ H(C) : sup exp(|z|)|f (z)| < ∞} z∈C

to conclude that D is hypercyclic. To show that D has a dense set of periodic, one has to prove that if P is the set of all functions eα (z) = exp(αz) such that there exists n ∈ N for which αn = 1, then P is dense in Ap (C). Corollary 6.4 The operator of differentiation D is hypercyclic and chaotic on the space Exp(C) of functions of exponential type.

7

Continuity of the Bergman Projection

We report on very recent work by Taskinen [56] and Jasiczak [34] about the continuity of the Bergman projection from a weighted space of measurable functions into a weighted space of holomorphic functions on the unit ball B of CN . We fix the notation. The scalar P N product in CN is hz, ξi = i=1 zi ξ. The modulus of a vector z is |z|2 = hz, zi. Setting r(z) := 1 − |z|2 we can write B = z ∈ CN : r(z) > 0

11

We denote by K(z, ξ) =

1 (1 − hz, ξi)N +1

the Bergman kernel. The Bergman projection is defined by B : L2 (B) −→ H 2 (B) R f 7→ Bf (z) := B f (ξ)K(z, ξ)dA(ξ). In the one dimensional case the Bergman projection B is a continuous operator from L2 (D) onto the Bloch space which properly contains the space H ∞ [58, Chapter 5]. The behavior of the Bergman projection on the unit ball of CN is similar. We consider the system V = (vn )n of weights on B, where each vn is defined by vn (z) = 1 if 0 ≤ |z| ≤ 1 − 1/e and vn (z) = | log(1 − |z|)|−n for 1 − 1/e ≤ |z| < 1. An easy calculation shows that vn ≥ vn+1 on B. Moreover lim −

|z|−→1

vn+1 (z) = 0. vn (z)

Therefore V H(B) := indn Hvn (B) is a weighted (LB)-space such that V H(B) = HV (B) holds topologically by Theorem 5.1 and H ∞ ⊂ V H(B). The associate system of weights is n V := v : B −→ ]0, ∞[ : continuous such that sup v(z) |log(1 − |z|)| < ∞ ∀n ∈ N . z∈B

We need the following weighted inductive limit of spaces of measurable functions V L∞ (B) := indL∞ vn (B), L∞ vn (B) := {f measurable : vn |f | essentially bounded on B} . The identity V L∞ (B) = L∞ V (B) holds algebraically and topologically by the arguments in [7]. Theorem 7.1 (Taskinen, Jasiczak [56, 34]) The Bergman projection B : V L∞ (B) −→ V H(B) is continuous. Proof. Let C > 0 and n ∈ N be such that |f | ≤ we have

C vn

almost everywhere on B. For ρ := |z|, Z

|B(f (z)| ≤ |f (ξ)||K(z, ξ)|dA(ξ) ≤ B Z Z C C ≤ |K(z, ξ)| dA(ξ) + |K(z, ξ)| dA(ξ) ≤ vn (|ξ|) vn (|ξ|) ρB B\ρB Z Z C C(1 − ρ)1/2 |K(z, ξ)| C0 ≤ |K(z, ξ)|dA(ξ) + dA(ξ) ≤ , 1/2 vn (ρ) B vn (ρ) vn+1 (|z|) B (1 − |ξ|) 12

since

Z |K(z, ξ)|dA(ξ) ≤ C0 | log(1 − |z|)| B

and

Z B

|K(z, ξ)| dA(ξ) ≤ C1 (1 − |z|)−1/2 . (1 − |ξ|)1/2 2

Theorem 7.2 (Taskinen, Jasiczak [56, 34]) Let E be a locally convex space such that L∞ (B) ⊂ E ⊂ L2 (B) which is endowed with a topology defined by weighted sup-seminorms. Let F be a topological subspace of E contained in H(B). If B : E −→ F is continuous then V L∞ (B) ⊂ E and V H(B) ⊂ F . Idea of the proof: One has to show that all the weights which define the topology of E (and F ) belong to V . Let w be a continuous weight on B which defines the topology of E. Since L∞ (B) ⊂ E, we have supz∈B w(z) < ∞. Using functions of the form (log(1−ζ))k ), k ∈ N, one shows that if supz∈B w(z)| log(1 − |z|)|k−1 w(z) < ∞, then supz∈B w(z)| log(1 − |z|)|k w(z) < ∞, see [34]. Consequently w ∈ V .

8

Sufficient sets and Sampling in A−∞

For f ∈ H(D), p > 0, and S ⊂ D, define kf kp,S = sup(1 − |z|)p |f (z)|. z∈S

The Banach space A

−p

= {f ∈ H(D) : kf kp,D < ∞} and the weighted inductive limit A−∞ = indp>0 A−p = ∪p>0 A−p

were considered in the Example 5.2 (1). Clearly A−p ⊂ A−q for p ≤ q. We set A−p (S) = f ∈ A−∞ : kf kp,S < ∞ . Definition 8.1 A subset S ⊂ D is called (p, q)-sampling (p ≤ q) if there exists C > 0 such that kf kq ≤ kf kp,S for all f ∈ A−q . For a set S ⊂ D, we define Σ(S) := {(p, q) ∈]0, ∞[×]0, ∞[: p ≤ q, S is (p, q) − sampling} . The set S is (p, p)-sampling if and only if S is sampling for A−p in the sense of Seip [50], i.e. if T : f −→ ((vp f )|S ) ∈ l∞ (S) is an isomorphism into. The (p, p)-sampling sets were characterized by Seip in [50] in terms of certain densities; see also [31]. Compare with Theorem 4.1. 13

Definition 8.2 (Horowitz, Korenblum, Pinchuk [32]) A subset S ⊂ D is called A−∞ sampling if the behavior of f on S determines the growth of f ∈ A−∞ . More precisely if n o −q inf q > 0 : f ∈ A = inf q > 0 : kf kq,S < ∞ . Definition 8.3 A subset S ⊂ D is weakly sufficient for A−∞ if for all p > 0, there exist q ≥ p and C > 0 such that kf kq ≤ C kf kp,S for all f ∈ A−∞ Sufficient sets for certain spaces of entire functions were introduced by Ehrenpreis in 1970 in connection with his Fundamental Principle. They were studied by Taylor in 1972 and Schneider in 1974. Napalkov and Korobeinik 1986 − 87 proved independently that sufficient sets and weakly sufficient sets coincide in a very general context. The definition above is equivalent to one of the conditions introduced by Schneider. Khoi and Thomas [36] studied in 1998 weakly sufficient sets for A−∞ . The references can be seen in [12]. Here is the relation between the aforementioned concepts. S is A−p -sampling for all p > 0 =⇒ S is A−∞ -sampling =⇒ S is weakly sufficient for A−∞ =⇒ S is a set of uniqueness for A−∞ . Moreover all the implications are strict. We report below on recent joint work with Doma´ nski [12]. Properties of (p, q)-sampling sets (1) For every S ⊂ D there exists a discrete subset S0 ⊂ S such that Σ(S) = Σ(S0 ). This is obtained as a consequence of a perturbation result. (2) If (p, q) ∈ Σ(S) then {(r, s) : r ≤ p, q − (p − r) ≤ s ≤ q} ⊂ Σ(S). (3) If (p, q) ∈ Σ(S) then Σ(S) 6= ∅.

p q , m m

∈ Σ(S) for each m ∈ N . In particular (0, 0) ∈ Σ(S) if

Theorem 8.4 The following are equivalent for S ⊂ D: (a) S is weakly sufficient for A−∞ . (b) R : A−∞ −→ indp>0 l∞ (vp , S), f 7→ f |S is an isomorphism into. (c) For all p > 0 there exists q ≥ p such that A−p (S) ⊂ A−q . (d) For all p > 0 there exists q ≥ p such that (p, r) ∈ Σ(S) for all r > q.

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Proof. The equivalence of (a) and (b) follows from Baernstein’s lemma, see [46, 26.26] and [48, 8.3.55]. The equivalence between (a) and (c) was proved by Khoi and Thomas [36]. The proof of the equivalence between (c) and (d) uses ideas of Seip [50]. 2 Examples 8.5

(1) For all p > 0 there exists S ⊂ D: Σ(S) : {(q, r) ∈ [0, p] × [0, p] : r ≥ q}.

This is based on results of Seip. (2) For each increasing sequence r = (rn )n such that r1 > 1/e and rn tends to 1 as n goes to ∞, the set E(r) :=

[

{z : |z| = rn }

n∈N

is (p, q)-sampling if and only if q − log rn sup − log(− log rn ) − log < ∞. p − log(− log rn ) n Here is an ingredient of the proof of (2): First observe that (1 − |z|2 )p ∼ (− log |z|)p for p > 0 near of |z| = 1. Set w(r) for the supremum of all log-convex functions such that. w(rn ) = (− log rn )p

n ∈ N.

Hadamard 3-circles theorem and a result of [14] permit to conclude that E(r) is (p, q)sampling if and only if sup w(r)(− log r)q < ∞. r∈]0,1[

This condition is evaluated further to reach the conclusion. We do not know a geometric characterization of arbitrary (p, q)-sampling sets.

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