Bergman spaces with exponential type weights

where Kz is the reproducing kernel of Aω . This condition allows us to obtain some interesting reproducing kernel estimates and more estimates on the solutions of the ∂-equation (Theorem 2.5) for more general weight ω∗. As an application, we prove the boundedness of the Bergman projection on Lω , identify the dual space of A p ω , and establish an atomic decomposition for it. Further, we give necessary and sufficient conditions for the boundedness and compactness of some operators acting from Aω into Aω , 1 ≤ p,q <∞, such as Toeplitz and (big) Hankel operators. MSC: 32A36; 30H35; 47B38; 41A65; 46J15


Introduction and results
Let H(D) denote the space of all analytic functions on D, where D is the open unit disk in the complex plane C. A weight is a positive function ω ∈ L 1 (D, dA), where dA(z) = dx dy π is the normalized area measure on D. For 0 < p < ∞, the weighted Bergman space A p (ω) consists of those functions f ∈ H(D) for which In this paper we study Bergman spaces with weights belonging to a certain class W, which we describe now. Decreasing weights ω satisfying conditions will be specified in what follows. The class W, considered previously in [24] and [12], consists of the radial decreasing weights of the form ω(z) = e -2ϕ(z) , where ϕ ∈ C 2 (D) is a radial function such that ( ϕ(z)) -1/2 τ (z) for some positive radial function τ (z) that decreases to 0 as |z| → 1and satisfies lim r→1 -τ (r) = 0. Here denotes the standard Laplace operator. Furthermore, we assume that either there exists a constant C > 0 such that τ (r)(1r) -C increases for r close to 1 or The prototype is the exponential weight (1.1) For the weights ω in our class, the point evaluations L z : f − → f (z) are bounded linear functionals on A 2 (ω) for each z ∈ D. In particular, the space A 2 (ω) is a reproducing kernel Hilbert space: for each z ∈ D, there are functions K z ∈ A 2 (ω) with L z = K z A 2 (ω) such that L z f = f (z) = f , K z ω , where f , g ω = D f (z)g(z)ω(z) dA(z).
The function K z is called the reproducing kernel for the Bergman space A 2 (ω) and has the property that K z (ξ ) = K ξ (z). The Bergman spaces with exponential type weights have attracted a lot of attention in recent years [9,12,16,17,24,25] since new techniques, different from those used for standard Bergman spaces, are required. Consider the class E that consists of the weights ω ∈ W satisfying D K z (ξ ) ω(ξ ) 1/2 dA(ξ ) ≤ Cω(z) -1/2 , z ∈ D.
(1.2) H g : A p ω p/2 − → L q ω q/2 that are bounded or compact. As mentioned earlier, one of the key tools consists in using the estimates for the ∂-equation obtained in Sect. 2.
In what follows we use the notation a b to indicate that there is a constant C > 0 with a ≤ Cb; and the notation a b means that a b and b a. Also, respectively the expressions L p ω and A p ω mean L p (D, ω p/2 dA) and A p (ω p/2 ) for 1lep < ∞.

Preliminaries and basic properties
A positive function τ on D is said to be of class L if it satisfies the following two properties: (A) There is a constant c 1 such that τ (z) ≤ c 1 (1 -|z|) for all z ∈ D; (B) There is a constant c 2 such that |τ (z)τ (ζ )| ≤ c 2 |zζ | for all z, ζ ∈ D. We also use the notation where c 1 and c 2 are the constants appearing in the previous definition. For a ∈ D and δ > 0, we use D(δτ (a)) to denote the Euclidean disc centered at a and having radius δτ (a). It is easy to see from conditions (A) and (B) (see [24,Lemma 2.1]) that if τ ∈ L and z ∈ D(δτ ((a)), then for sufficiently small δ > 0, that is, for δ ∈ (0, m τ ). This fact will be used many times in this paper.
Definition 2. 1 We say that a weight ω is of class L * if it is of the form ω = e -2ϕ , where ϕ ∈ C 2 (D) with ϕ > 0 and ( ϕ(z)) -1/2 τ (z) with τ (z) is a function in the class L.
It is straightforward to see that W ⊂ L * . The following result is from [24, Lemma 2.2] and gives the boundedness of the point evaluation functional on A 2 ω .
Lemma A Let ω ∈ L * , 0 < p < ∞, and z ∈ D. If β ∈ R, there exists M ≥ 1 such that for all f ∈ H(D) and all sufficiently small δ > 0.
We also need a similar estimate for the gradient of |f |ω 1/2 .
The following lemma on coverings is due to Oleinik, see [21].
Lemma B Let τ be a positive function on D of class L, and let δ ∈ (0, m τ ). Then there exists a sequence of points {z n } ⊂ D such that the following conditions are satisfied: The multiplicity N in Lemma B is independent of δ, and it is easy to see that one can take, for example, N = 256. Any sequence satisfying the conditions in Lemma B will be called a (δ, τ )-lattice. Note that |z n | → 1as n → ∞. In what follows, the sequence {z n } will always refer to the sequence chosen in Lemma B.

Integral estimates for reproducing kernels
We use the notation k z for the normalized reproducing kernels in A p ω , that is, The next result (see [4,17,24] for (a) when p = 2 and [18, Lemma 3.6] for part (b)) provides useful estimates involving reproducing kernels.
Theorem A Let K z be the reproducing kernel of A 2 ω . Then (a) For ω ∈ W, one has (2.7) (b) For all sufficiently small δ ∈ (0, m τ ) and ω ∈ W, one has Proof By Lemma A and condition (1.2), we have which finishes the proof. Lemma 2.3 together with condition (1.2) allows us to obtain the following estimate for the norm of the reproducing kernel in A p ω . Lemma 2.4 Let 1 ≤ p < ∞, ω ∈ E, and z ∈ D. Then Proof By (1.2) and Lemma 2.3, we have On the other hand, by using statement (b) of Theorem A, we have This completes the proof.

Estimates for the ∂-equation
The following result, which provides more estimates on the solutions of the ∂-equation, will play a crucial role in describing the bounded Hankel operators acting from A p ω to A q ω when 1 ≤ p ≤ q < ∞. Also, it can be of independent interest.

Theorem 2.5
Let ω ∈ W, and consider the associated weight ω * (z) := ω(z)τ (z) α , z ∈ D, and α ∈ R. Then there exists a solution u of the equation ∂u = f such that for all 1 ≤ p < ∞, provided the right-hand side is finite. Moreover, one also has the L ∞estimate Proof We follow the method used in [5] where the case α = 0 was proved. By Lemma 3.1 in [24], there are holomorphic functions F a and some δ 0 ∈ (0, m τ ) such that (2.9) Let δ 1 < δ 0 . Then there is a sequence {z n } n≥1 such that {D(δ 1 τ (z n ))} is a covering of D of finite multiplicity N and satisfies the other statements of Lemma B. Let χ n be a partition of unity subordinate to the covering D(δ 1 τ (z n )). Consider Since F z n are holomorphic functions on D, by the Cauchy-Pompeiu formula, we have Since χ n is a partition of the unity, we have On the other hand, assume that Then, by (2.13), it is straightforward that the L ∞ -estimate holds. Our next goal is to prove the inequality Consider g(ξ ) := f (ξ )ω * (ξ ) 1/2 and Tg(z) := D G(z, ξ )g(ξ ) dA(ξ ). Then the last inequality takes the form Therefore, using Hölder's inequality and (2.13), we have These estimates and Fubini's theorem give Now, using the expression of the kernel G(z, ξ ) and the fact that χ n are supported in By (2.11) and using the fact that where the last inequality above follows from the fact that {D(δ τ (z n ))} is a covering of D of finite multiplicity. Now we are going to prove that First we consider the covering of {ξ ∈ D : |zξ | > δ 2 τ (z)} given by Let 4δ 1 < δ 2 < δ 0 5 and z ∈ D be fixed. If ξ ∈ D(δ 2 τ (z)) ∩ D(δ 1 τ (z n )), using (2.1), we have that implies z ∈ D(δ 0 τ (z n )). Using (2.1) and property (i) of (2.9), it follows Therefore, using again (2.1) and polar coordinates, we get this implies a contradiction with our assumption. Thus, Then, again using τ (ξ ) τ (z n ) and property (ii) of (2.9) with • If 2 + α ≤ 0. Using condition (B) in the definition of the class L, it follows that This together with (2.14) establishes (2.13).

Carleson type measures
We are going to define (vanishing) q-Carleson measures for A p ω , 0 < p, q < ∞, for weights ω in the class W and give some essential theorems.

Definition 2.6
Given ω ∈ W and 0 < p, q ≤ ∞, let μ be a positive measure on D. We say that μ is a q- Next, the following theorems were essentially proved in [24,Theorem 1]. They established necessary and sufficient conditions for I μ : A p ω − → L q (D, dμ) to be bounded (compact) when 0 < p, q < ∞.

Theorem B
Given ω ∈ W and 0 < p ≤ q < ∞, let μ be a finite positive Borel measure on D.
Theorem C Given ω ∈ W and 0 < p ≤ q < ∞, let μ be a finite positive Borel measure on D.
Theorem D Given ω ∈ W and 0 < q < p < ∞, let μ be a finite positive Borel measure on D.
The following conditions are equivalent:

Bounded projections
The boundedness of Bergman projection is a fact of fundamental importance. In the case of the unit disc, the boundedness of Bergman projections is studied in [13,38], and it immediately gives the duality between the Bergman spaces. The natural Bergman projection is not necessarily bounded on L p ω unless p = 2 (see [8] and [35] for more details). However, we are going to see next that P ω is bounded on L p ω for weights ω in the class E. We first prove the boundedness of the sublinear operator P + ω defined as We mention here that, for the case of the exponential weight with σ = 1, the results of this section have been obtained recently in [7]. Proof We first consider the easiest case p = 1. By Fubini's theorem and condition (1.2), we obtain Next, we consider the case 1 < p < ∞. Let p denote the conjugate exponent of p. By Hölder's inequality and (1.2), we get This together with Fubini's theorem and another application of (1.2) gives Finally, if f ∈ L ∞ (ω 1/2 ), by condition (1.2) we get This shows that P + ω is bounded on L ∞ (ω 1/2 ). The proof is complete.  Proof This is an immediate consequence of the boundedness of the Bergman projection and the density of the polynomials.

Duality
As in the case of the standard Bergman spaces, one can use the result just proved on the boundedness of the Bergman projection P ω in L p ω to identify the dual spaces of A p ω . As usual, if X is a Banach space, we denote its dual by X * . Next two results (Theorems 4.1 and 4.2) on the duality of Bergman spaces with exponential type weights appear also on [7].
Here p denotes the conjugate exponent of p, that is, p = p/(p -1).
Proof Let 1 < p < ∞ and let p = p/(p -1) be its dual exponent. Given a function g ∈ A p ω , Hölder's inequality implies that the linear functional ψ g : Conversely, let T ∈ (A p ω ) * . By the Hahn-Banach theorem, we can extend T to an element T ∈ (L p ω ) * such that T = T . By the Riesz representation theorem, there exists H ∈ L p (D, ω p/2 dA) with H L p (ω p/2 ) = T = T such that Let g = P ω h. By Theorem 3.2, the Bergman projection P ω : From Fubini's theorem it is easy to see that P ω is self-adjoint. Indeed, The interchange of the order of integration is well justified, because of the boundedness of the operator P + ω (see Theorem 3.1) given by Therefore, since f = P ω f for every f ∈ A p ω , according to Corollary 3.3, we get Finally, the function g is unique. Indeed, if there is another function g ∈ A p ω with T(f ) = ψ g (f ) = ψ g (f ) for every f ∈ A p ω , then by taking f = K a for each a ∈ D (that belongs to A p ω due to Lemma 2.4) and using the reproducing formula, we obtain g(a) = ψ g (K a ) = ψ g (K a ) = g(a), a ∈ D.
Thus, any bounded linear functional T is of the form T = ψ g for some unique g ∈ A p ω and, furthermore, The proof is complete.

Theorem 4.2
Let ω ∈ E. The dual space of A 1 ω can be identified (with equivalent norms) with A ∞ (ω 1/2 ) under the integral pairing f , g ω .
Conversely, let T ∈ (A 1 ω ) * . Consider the space X that consists of the functions of the form h = f ω 1/2 with f ∈ A 1 ω . Clearly, X is a subspace of L 1 (D, dA) and F(h) := T(hω -1/2 ) = T(f ) defines a bounded linear functional on X with F = T . By the Hahn-Banach theorem, F has an extension F ∈ (L 1 (D, dA)) * with F = F . Hence, there is a function G ∈ L ∞ (D, dA) with G L ∞ (D,dA) = F such that or Also, for f ∈ A 1 ω , by the reproducing formula, we have Finally, as in the proof of Theorem 4.1, the function g is unique.

Corollary 4.3 Let ω ∈ E. The set E of finite linear combinations of reproducing kernels is dense in
Proof Since E is a linear subspace of A p ω , by standard functional analysis and the duality results in Theorems 4.1 and 4.2, it is enough to prove that g ≡ 0 if g ∈ A p ω satisfies f , g ω = 0 for each f in E (with p being the conjugate exponent of p, and g ∈ A ∞ (ω 1/2 ) if p = 1). But, taking f = K z for each z ∈ D and using the reproducing formula, we get g(z) = P ω g(z) = g, K z ω = 0 for each z ∈ D. This finishes the proof.
Our next goal is to identify the predual of A 1 ω . For a given weight v, we introduce the

Theorem 4.4 Let ω ∈ E.
Under the integral pairing f , g ω , the dual space of A 0 (ω 1/2 ) can be identified (with equivalent norms) with A 1 ω .
Proof If g ∈ A 1 ω , clearly g (f ) = f , g ω defines a bounded linear functional in A 0 (ω 1/2 ) with g ≤ g A 1 ω . Conversely, assume that ∈ (A 0 (ω 1/2 )) * . Consider the space X that consists of functions of the form h = f ω 1/2 with f ∈ A 0 (ω 1/2 ). Clearly, X is a subspace of C 0 (D) (the space of all continuous functions vanishing at the boundary) and T(h) = (ω -1/2 h) = (f ) defines a bounded linear functional on X with T = . By the Hahn-Banach theorem, T has an extension T ∈ (C 0 (D)) * with T = T . Hence, by the Riesz representation theorem, there is a measure μ ∈ M(D) (the Banach space of all complex Borel measures μ equipped with the variation norm μ M ) with μ M = T such that Consider the function g defined on the unit disk by Clearly, g is analytic on D and, by Fubini's theorem and condition (1.2), we have proving that g belongs to A 1 ω . Now, since A 0 (ω 1/2 ) ⊂ A 2 ω , the reproducing formula f (ζ ) = f , K ζ ω holds for all f ∈ A 0 (ω 1/2 ). This and Fubini's theorem yield By the reproducing formula, the function g is uniquely determined by the identity g(z) = (K z ). This completes the proof.
For the case of normal weights, the analogues of Theorems 4.2 and 4.4 were obtained by Shields and Williams in [32]. They also asked what happens with the exponential weights, problem that is solved in the present paper.

Atomic decomposition
For 1 ≤ p < ∞, in this section we are going to obtain an atomic decomposition for the large weighted Bergman space A p ω , that is, we show that every function in the Bergman spaces A p ω with ω in the class E can be decomposed into a series of kernel functions. With the help of the duality results and the estimates for the p-norm of the reproducing kernels K z , we can reach our goal. Before stating the main theorem of this section, we need two auxiliary lemmas as follows.
We may rephrase Lemma 5.1 by saying that every (ε, τ )-lattice, with ε > 0 small enough, is a sampling sequence for the Bergman space A p ω . Recall that {z k } ⊂ D is a sampling sequence for the Bergman space for any f ∈ A p ω . Just note that Lemma 5.1 gives one inequality, and the other follows by standard methods using Lemma A and the lattice properties. Sampling sequences on the classical Bergman space were characterized by K. Seip [30] (see also the monographs [10] and [31]). For sampling sequences on large weighted Bergman spaces, we refer to [4].
The function given by Proof By Hölder's inequality, Lemma A, and Lemma 2.4, it is easy to see that the partial sums of the series in (5.1) converge uniformly on compact subsets of D. Thus, F defines an analytic function on D. Furthermore, for p = 1, using (1.2) we have For the case p > 1, consider By Hölder's inequality, we have On the other hand, using Lemma A, Lemma B, and (1.2), we have Therefore, applying again (1.2), we obtain which completes the proof.
Now we are ready to state our main result related to an atomic decomposition of large weighted Bergman spaces A p ω for 1 ≤ p < ∞. Recall that k p,z is the normalized reproducing kernel in A p ω , that is,

Theorem 5.3
Let ω ∈ E and 1 ≤ p < ∞. There exists a τ -lattice {z n } ⊂ D such that: (i) For any λ = {λ n } ∈ p , the function Proof On the one hand, statement (i) is exactly Lemma 5.2. On the other hand, in order to prove (ii), we define a linear operator S : p − → A p ω given by λ n k p,z n .
By (i), the operator S is bounded. By the duality results obtained in the previous section, when 1 < p < ∞, the adjoint operator S * : A p ω → p , where p is the conjugate exponent of p, is defined by x n S * f n for every x ∈ p and f ∈ A p ω . To compute S * , let e n denote the vector that equals 1 at the nth coordinate and equals 0 at the other coordinates. Then Se n = k p,z n , and using the reproducing formula, we get Hence, S * : A p ω − → p is given by We must prove that S is surjective in order to finish the proof of this case. By a classical result in functional analysis, it is enough to show that S * is bounded below. By Lemma 5.1 and Lemma 2.4, we obtain which shows that S * is bounded below. Finally, once the surjectivity is proved, the estimate is a standard application of the open mapping theorem. When p = 1, then Hence we must show that for f ∈ A ∞ (ω 1/2 ). However, this can be proved with the same method as Lemma 5.1. Indeed, let z ∈ D. Then there is a point z n with z ∈ D(ετ (z n )). By Lemma A, we have As done in the proof of Lemma 5.1, we have Thus, putting this in the previous estimate, we obtain Finally, taking the supremum on z and ε > 0 small enough so that C 4 ε ≤ 1/2, we have The proof is complete.

Toeplitz operators
In this section we are going to extend the results given in [3, Theorem 1.1] to the non-Hilbert space setting, when the weight ω is in the class E. Concretely, we characterize the bounded and compact operators T μ acting from A p ω to A q ω when 1 ≤ p, q < ∞. Recall that the Toeplitz operator T μ is defined by Note that T μ is very loosely defined here, because it is not clear when the integrals above will converge, even if the measure μ is finite. We suppose that μ is a finite positive Borel measure that satisfies the condition Then the Toeplitz operator T μ is well defined on a dense subset of A p ω , 1 ≤ p < ∞. In fact, by Corollary 4.3 and Theorem 5.3, the set E of finite linear combinations of reproducing kernels is dense in A p ω . Therefore, it follows from condition (6.1) and the Cauchy-Schwarz inequality that T μ (f ) is well defined for any f ∈ E. Also, recall that, for δ ∈ (0, m τ ), the averaging function of μ on D is given by Moreover, Proof Since we have the estimate K z A p ω ω(z) -1/2 τ (z) -2(p-1)/p , if we assume that the Toeplitz operator T μ : A p ω − → A q ω is bounded, then we obtain (6.2) with the same argument as in the proof of Theorem 1.1 in [3].
Conversely, we suppose that (6.2) holds. We first prove that Indeed, by Lemma A, we have Then, by Fubini's theorem, the fact that τ (s) τ (ξ ) for s ∈ D(δτ (ξ )), and condition (1.2), we get This establishes (6.3). Now we proceed to prove that T μ is bounded. If q > 1, by Hölder's inequality, we obtain Using (6.3), we have If q = 1, this holds directly. By Fubini's theorem and condition (1.2), we obtain Consider the measure ν given by Since (6.2) holds, by Theorem B, the identity I ν : A p ω − → L q (D, dν) is bounded. Moreover, I ν E(μ) 1/q . Therefore, This finishes the proof.
For each sufficiently small δ > 0, μ δ ∈ L pq p-q (D, dA). Moreover, . Proof (i) ⇒ (ii) For an arbitrary sequence λ = {λ k } ∈ p , we consider the function where r k (t) is a sequence of Rademacher functions (see [20] or Appendix A of [11] ) and {z k } is the sequence given in Lemma B. Because of the norm estimate given in Lemma 2.4, by part (i) of Theorem 5.3, we obtain In other words, we have Integrating with respect to t from 0 to 1, applying Fubini's theorem, and invoking Khinchine's inequality (see [20]), we obtain Let χ k denote the characteristic function of the set D(3δτ (z k )). Since the covering {D(3δτ (z k ))} of D has finite multiplicity N , we have On the other hand, since for small δ > 0 we have |K z k (z)| K z k A 2 ω K z A 2 ω for every z ∈ D(δτ (z k )), applying statement (a) of Theorem A and (2.1), we have That is, Then, using the duality between p/q and p p-q , we conclude that This is the discrete version of our condition. To obtain the continuous version, simply note that This finishes the proof of this implication.
(ii) ⇒ (i) First we begin with the easiest case q = 1. By Fubini's theorem and condition (1.2), we have Now, by using Theorem D with the measure given by we obtain the desired result. Finally, we study the case 1 < q < ∞. Let {z j } be the sequence given in Lemma B. Applying Lemma A and Lemma B, we obtain Applying Hölder's inequality, we get Thus This gives which, by Fubini's theorem and condition (1.2), gives Combining this with using (2.1) and Proposition 6.2 shows that This proves the desired result.
Next we characterize compact Toeplitz operators on weighted Bergman spaces A p ω for weights ω in the class E. We need first a lemma. Lemma 6.4 Let 1 < p < ∞, and let k p,z be the normalized reproducing kernels in A p ω , with ω ∈ E. Then k p,z → 0 weakly in A p ω as |z| → 1 -.
Proof By duality and the reproducing kernel properties, we must show that |g(z)|/ K z A p ω goes to zero as |z| → 1whenever g is in A p ω , where p denotes the conjugate exponent of p, but this follows easily by the density of the polynomials and Lemma A. For each z ∈ D, consider the function g z (ξ ) := (g(z)g(ξ ))K z (ξ ). Condition (7.1) ensures that g z ∈ A 1 ω , and by the reproducing formula in Corollary 3.3, one has H g K a (z) = D g(z)g(ξ ) K a (ξ )K z (ξ )ω(ξ ) dA(ξ ) = g z , K a ω = g z (a). Now, for δ small enough, we have |K z (a)| K z A 2 ω K a A 2 ω for z ∈ D(δτ (a)). Hence, by the statement (a) of Theorem A and (2.1), we have q ω D f (z) q g (z) q ω(z) q/2 τ (z) q dA(z).

(7.3)
Proof Suppose that τ g ∈ L r (D, dA). By (7.3), since p/q > 1, a simple application of Hölder's inequality, yields This proves the boundedness of H g . Conversely, pick ε > 0 and let {z k } be a (ε, τ )-lattice on D. For a sequence λ = {λ k } ∈ p , we consider the function where r k (t) is a sequence of Rademacher functions. Because of the norm estimate for reproducing kernels given in Lemma 2.4, by part (i) of Theorem 5.3, we obtain Thus, the boundedness of H g gives Therefore, D ∞ k=1 λ k r k (t)ω(z k ) 1/2 τ (z k ) 2( p-1 p ) H g K z k (z) q ω(z) q/2 dA(z) λ q p .