Reproducing Kernel Estimates, Bounded Projections and Duality on Large Weighted Bergman Spaces

We obtain certain estimates for the reproducing kernels of large weighted Bergman spaces. Applications of these estimates to boundedness of the Bergman projection on Lp(D,ωp/2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p({\mathbb {D}},\omega ^{p/2})$$\end{document}, complex interpolation and duality of weighted Bergman spaces are given.


Introduction
Let D = {z ∈ C : |z| < 1} be the unit disk in the complex plane C, dA(z) = dxdy π be the normalized area measure on D, and let H (D) denote the space of all analytic functions on D. A weight is a positive function ω ∈ L 1 (D, dA). When ω(z) = ω(|z|) for all z ∈ D, we say that ω is radial.
For 0 < p < ∞, the weighted Bergman space A p (ω) is the space of all functions f ∈ H (D) such that Communicated by Richard Rochberg.
Our main goal is to study certain properties of the Bergman spaces A p (ω) for a large class of weights, including certain rapidly radial decreasing weights, that is, weights that are going to decrease faster than any standard weight (1 − |z| 2 ) α , α > 0, such as the exponential-type weights (1.1) For the weights ω considered in this paper, for each z ∈ D the point evaluations L z are bounded linear functionals on A p (ω). In particular, the space A 2 (ω) is a reproducing kernel Hilbert space: for each z ∈ D, there are functions is the natural inner product in L 2 (D, ωdA). The function K z has the property that K z (ξ ) = K ξ (z), and is called the reproducing kernel for the Bergman space A 2 (ω).
It is straightforward to see from the previous formula that the orthogonal (Bergman) projection from L 2 (D, ωdA) to A 2 (ω) is given by Some basic properties of the Bergman spaces with radial rapidly decreasing weights are not yet well understood and have attracted some attention in recent years [1,6,7,9,11,17]. The interest in such spaces arises from the fact that the usual techniques for the standard Bergman spaces fail to work in this context, and therefore new tools must be developed. For example, the natural Bergman projection P ω is not necessarily bounded on L p (D, ωdA) unless p = 2 (see [5] for the exponential weights and [20] for more examples). Another difficulty when studying these spaces arises from the lack of an explicit expression of the reproducing kernels. It turns out that when studying properties or operators such as the Bergman projection where the reproducing kernels are involved, the most convenient settings are the spaces A p (ω p/2 ) (or the weighted Lebesgue spaces L p (ω p/2 ) := L p (D, ω p/2 dA)), and our first main result is that, for the class of weights ω considered, the Bergman projection P ω is bounded from L p (ω p/2 ) to A p (ω p/2 ) for 1 ≤ p < ∞ (see Theorem 4.1). A consequence of that result will be the identification of the dual space of A p (ω p/2 ) with the space A p (ω p /2 ) under the natural integral pairing , ω , where p denotes the conjugate exponent of p.
The key ingredient for obtaining the previously mentioned results is a certain integral-type estimate involving the reproducing kernels K z . This integral estimate will be deduced from a pointwise estimate for |K z (ξ )| that can be of independent interest (Theorem 3.1). The pointwise estimate obtained can be thought as the analogue for weighted Bergman spaces of the corresponding ones obtained by Marzo and Ortega-Cerdà [13] for reproducing kernels of weighted Fock spaces.
Throughout this work, the letter C will denote an absolute constant whose value may change at different occurrences. We also 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.

Preliminaries
In this section we provide the basic tools for the proofs of the main results of the paper. For a ∈ D and δ > 0, we use the notation D(δτ (a)) for the Euclidean disk centered at a and radius δτ (a).
A positive function τ on D is said to belong to the class L if it satisfies the following two properties: We also use the notation where c 1 and c 2 are the constants appearing in the previous definition. It is easy to see from conditions (A) and (B) (see [16,Lemma 2.1]) that if τ ∈ L and z ∈ D(δτ (a)), for sufficiently small δ > 0, that is, for δ ∈ (0, m τ ). This fact will be used several times in this work.
It can be seen from the proof given in [16] that one only needs f to be holomorphic in a neighborhood of D(δτ (z)). Another consequence of the above result is that the Bergman space A p (ω β ) is a Banach space when 1 ≤ p < ∞ and a complete metric space when 0 < p < 1.
Since the norm of the point evaluation functional equals the norm of the reproducing kernel in A 2 (ω), the result of Lemma A also gives an upper bound for K z A 2 (ω) . The next result [10] says that (at least for some class of weights) this upper bound is the corresponding growth of the reproducing kernel.

Lemma B
Let ω ∈ L * and suppose that the associated function τ (z) also satisfies the condition The result in Lemma B has also been obtained [3,16] for radial weights ω ∈ L * for which the associated function τ (|z|) = τ (z) decreases to 0 as r = |z| → 1 − , τ (r ) → 0 as r → 1, and moreover, either there exists a constant C > 0 such that τ (r )(1 − r ) −C increases for r close to 1 or lim r →1 − τ (r ) log 1 τ (r ) = 0. If a function τ satisfies the condition (C), it does not necessarily hold that kτ (z) satisfies the same condition (C) for all k > 0 (an example of this phenomenon are the standard weights (1 − |z| 2 ) β ), but this is true for the exponential-type weights given by (1.1), and therefore these weights satisfy the following strongest condition (D) For each m ≥ 1, there are constants b m > 0 and 0 < t m < 1/m such that This leads us to the following definition. Definition 2.2 A weight ω is in the class E if ω ∈ L * and its associated function τ satisfies the condition (D).
The prototype of a weight in the class E are the exponential-type weights given by (1.1). An example of a non-radial weight in the class E is given by where p > 0, ω is a radial weight in the class E, and f is a non-vanishing analytic function in A p (ω).
In order to obtain pointwise estimates for the reproducing kernels, we will need the classical Hörmander's theorem [8, provided the right-hand side integral is finite.
Recall that the operators ∂ and ∂ are defined by provided the use of the identification z = x + iy is made. Also = 4∂∂.
and for any g one can find v such that ∂v = g with then for the solution v 0 with minimal norm in L 2 (D, e −2ϕ dA), one has where C = 6/(1 − C 1 ).

Estimates for Reproducing Kernels
In this section we will give some pointwise estimates for the reproducing kernel, especially far from the diagonal, as well as an integral-type estimate involving the reproducing kernel. For weights in the class L * , and points close to the diagonal, one has the following well-known estimate (see [11,Lemma 3.6], for example) for all δ ∈ (0, m τ ) sufficiently small. Thus, the interest of the next result relies when we are far from the diagonal.
The proof is going to be similar to the pointwise estimate obtained by Marzo and Ortega-Cerdà in [13] for weighted Fock spaces. However, if one wants to follow the proof given in [13] one needs the function τ to satisfy the following condition: for all z, ξ ∈ D with z / ∈ D(δτ (ξ )) there is η > 0 sufficiently small and a constant c 3 > 0 such that and it is easy to see that this condition is never satisfied in the setting of the unit disk. Instead of that, we will use a mixture of the arguments in [13] with the ones in [15]. Before going to the proof of Theorem 3.1, we need an auxiliary result.
Proof Since ϕ(z) τ (z) −2 and ε > 0 is arbitrary, it suffices to show that An easy computation gives Another computation yields where c 2 is the constant appearing in condition (B). Since clearly we only need to choose β big enough satisfying Also, Hence choosing β big enough so that Thus, to get both (3.2) and (3.3) we only need to choose β > 0 satisfying Here t m is the number appearing in condition (D). On the other hand, Thus, we only need which gives (3.4), finishing the proof.
due to (2.1), and therefore the result follows from the inequality (3.5) where P ω is the Bergman projection given by (1.2), which is obviously bounded from Consider the subharmonic function with β > 0 (depending on M) being the one given in Lemma 3.2, which gives We will write τ ϕ (z) if we need to stress the dependence on ϕ. We consider the function Notice that the function u is holomorphic in D(δτ ρ (z)) for some δ > 0. Then, using the notation ω ρ = e −2ρ , by Lemma A and the remark following it, We know that (ϕ + ϕ ξ ) > 0; then applying Hörmander's Theorem A, one has v such This together with (3.7) says that we are under the assumptions of Berndtsson's theorem. Therefore, since u is the solution with minimal norm in L 2 (D, e −2ϕ dA) of the equation (3.6), using Theorem B we obtain Putting these into (3.9), using (3.6), (3.8), and the fact that Clearly, the function ϕ ξ is bounded in D(δτ (ξ )), and therefore Thus, we obtain This gives Then, taking into account (3.5), we get Finally, interchanging the roles of z and ξ we also get completing the proof of the theorem.

Lemma 3.3 Let ω ∈ E and let K z be the reproducing kernel for A 2 (ω). Then there exists a constant C > 0 such that
By Lemma B and (2.1), On the other hand, by Theorem 3.1 with M = 3, we have (3.11) To estimate the last integral, consider the covering of {ξ ∈ D : |z − ξ | > δ 0 τ (z)} given by Putting this into (3.11) we get which together with (3.10) gives the desired result.

Bounded Projections and the Reproducing Formula
Recall that the natural Bergman projection P ω is given by As was said in the Introduction, the Bergman projection is not necessarily bounded on L p (D, ω dA) unless p = 2. However, we are going to see next that P ω is bounded on The Bergman projection P ω : Proof We first consider the easiest case p = 1. By Fubini's theorem and Lemma 3.3 we obtain Next, we consider the case 1 < p < ∞. Let p denote the conjugate exponent of p. By Hölder's inequality and Lemma 3.3, This together with Fubini's theorem and another application of Lemma 3. 3 gives The proof is complete.
To deal with the case p = ∞, given a weight v, we introduce the growth space L ∞ (v) that consists of those measurable functions f on D such that and let A ∞ (v) be the space of all analytic functions in L ∞ (v).

Theorem 4.2
Let ω ∈ E. The Bergman projection P ω : This shows that P ω is bounded.
As a consequence of the results obtained on bounded projections, we obtain the following result.

Corollary 4.3
Let ω ∈ E. The following are equivalent: Proof By the definition of the projection P ω and the properties of the reproducing kernel, we always have f = P ω f for every f ∈ A 2 (ω). Thus (i) is easily implied by the density condition in (ii) and the boundedness of P ω in L 1 (ω 1/2 ). Conversely, suppose that (i) holds and let f ∈ A 1 (ω 1/2 ). Since L 2 (ω) is dense in L 1 (ω 1/2 ), we can find functions g n ∈ L 2 (ω) with f − g n L 1 (ω 1/2 ) → 0. Set f n = P ω g n ∈ A 2 (ω). Then, by (i) and Theorem 4.1, we have The proof is complete.
The identity f = P ω f appearing in (i) is usually referred to as the reproducing formula. If the weight ω is radial, then the polynomials are dense in A 1 (ω 1/2 ) and thus the reproducing formula in (i) holds. The fact that the reproducing formula also holds for non-radial weights in the class E is not obvious, and it is our goal to establish that result in the next subsections.

Associated Weighted Bergman Spaces
We need to consider reproducing kernels K * z of the Bergman space A 2 (ω * ), where the weight ω * is of the form for all f ∈ H (D) and all δ > 0 sufficiently small.
Proof This is an immediate consequence of Lemma A and (2.1). Indeed, This finishes the proof.
As in Lemma A, it suffices that f be holomorphic in a neighborhood of D(δτ (z)) to get the conclusion in Lemma 4.4. As a consequence, we get the estimate K * z 2 We also need the analogue of Theorem 3.1 for the reproducing kernels K * z . Since we do not know if ω * belongs to the class E, we cannot deduce the result from Theorem 3.1, so we must repeat the proof with appropriate modifications. Before doing that, we need to establish more estimates of the solutions of the ∂-equation, a result that can be of independent interest.
For each j define Since h a j (z) is holomorphic on D, by the Cauchy-Pompeiu formula we have ∂(S j f ) = f χ j , and therefore ζ ) is given by translates to the estimate
Since δ 1 /δ 0 ≤ 1/2, we obtain |z − ζ | ≤ C|z − a j |, which together with the fact that τ (ζ ) τ (a j ) for ζ ∈ D(δ 1 τ (a j )) yields Therefore, since there are at most N points a j with ζ ∈ D(δ 1 τ (a j )), This gives where the last inequality is proved in a manner similar to that in the proof of Lemma 3.3, but in the case α < 0 one must use that τ (ζ ) 2 k τ (z) for ζ ∈ R k (z) (a consequence of condition (B) in the definition of the class L), where R k (z) are the same sets used in the proof of Lemma 3.3. This together with (4.4) establishes (4.3).
Using (4.3), it is straightforward to see that the L ∞ -estimate holds. Now, let 1 ≤ p < ∞. By Hölder's inequality and (4.3), This, Fubini's theorem, and property (a) gives We handle the inside integral in a similar form as done before. By property (a), for ζ ∈ D(δ 1 τ (a j )), we have The integral from outside the disk D(δ 0 τ (a j )) is estimated with the same method as done in the proof of (4.3) using property (b), so that we obtain Putting this into the previous estimate, we finally obtain since {D(δ 1 τ (a j ))} is a covering of D of finite multiplicity. This proves (4.2), completing the proof of the proposition. Now we can prove the analogue of Theorem 3.1 for the reproducing kernels K * z .

Lemma 4.6
Let ω ∈ E, and K * z be the reproducing kernel of A 2 (ω * ), where ω * is the associated weight given by (4.1). For each M ≥ 1, there exists a constant C > 0 (depending on M) such that for each z, ξ ∈ D one has Proof Let z, ξ ∈ D and fix 0 < δ < m τ . The result is clear if D(δτ (z)) ∩ D(δτ (ξ )) = ∅, so that we assume D(δτ (z)) ∩ D(δτ (ξ )) = ∅. Let 0 ≤ χ ≤ 1 be a function in C ∞ (D) with compact support in the disk D(δτ (ξ )) such that χ ≡ 1 in D( δ 2 τ (ξ)) and |∂χ| 2 χ τ (ξ) 2 . By Lemma 4.4 we obtain (4.5) By duality, K * z L 2 (D,χ ω * ) = sup f | f, K * z L 2 (D,χ ω * dA) |, where the supremum runs over all holomorphic functions f on D(δτ (ξ )) such that where P ω * is the orthogonal Bergman projection, which is obviously bounded from L 2 (D, ω * dA) to A 2 (ω * ). Now we consider u = f χ − P ω * ( f χ) the solution with minimal norm in L 2 (D, ω * dA) of the equation (4.7) Since For a given 0 < ε < 1/2, consider the subharmonic function from Lemma 3.2 given by with β > 0 (depending on M and ε) taken big enough so that Thus ϕ ≤ (ϕ + ϕ ξ ) ≤ 2 ϕ and 1 2 ϕ ≤ (ϕ − ϕ ξ ) ≤ ϕ. Next, we are going to apply the method used in the proof of Berndtsson's theorem. Since u is the solution with minimal norm in L 2 (D, ω * dA) of the equation (4.7), it satisfies u, h ω * = 0 for any square integrable holomorphic function h in D. This clearly implies that D u 0 h ω * e −2ϕ ξ dA = 0 for any such h, with u 0 = u e 2ϕ ξ . Thus u 0 is the minimal solution in L 2 (D, ω * e −2ϕ ξ dA) of the equation ∂v = ∂(u e 2ϕ ξ ) := g. By Proposition 4.5 applied with the weight ω ξ = ωe −2ϕ ξ , we can find a solution v of the equation ∂v = g satisfying Hence the same estimate is true for the minimal solution u 0 , which implies Now use (4.8) with ε > 0 taken so that Cε ≤ 1/2, and absorb the last member of the right-hand side in the left-hand side. The result is Arguing as in the proof of Lemma 4.4, then applying (4.9), we obtain Since the function ϕ ξ is bounded in D(δτ (ξ )), this and (4.6) yields Thus, taking into account (4.5), we get Finally, interchanging the roles of z and ξ we obtain the desired result.

Corollary 4.7
Let ω ∈ E, and K * z be the reproducing kernel for A 2 (ω * ), where ω * is the associated weight given by (4.1). For β ∈ R, there exists a constant C > 0 such that Proof Apart from the extra factor τ (z) β , this is almost the analogue of Lemma 3.3. For the proof, just use the same method applying Lemma 4.6 with M taken big enough, but in the case β − 1 > 0, use that τ (ξ) 2 k τ (z) for z ∈ R k (z).
Arguing in the same way as in the proof of the boundedness of the Bergman projection, using Corollary 4.7 with β = 0, we can prove that P ω * is bounded on L p (ω p/2 * ), but in order to obtain the reproducing formula, what is really needed is the following result.

Lemma 4.8
Let ω ∈ E, 1 ≤ p < ∞ and let ω * be the associated weight given by (4.1). Then P ω * : Proof This is proved with the same method used in the proof of Theorem 4.1, but using Corollary 4.7 instead of Lemma 3.3. We leave the details to the interested reader.

The Reproducing Formula
With all the machinery developed in the previous subsection, we can prove the following key result, from which the reproducing formula will follow.
Proof Our proof has its roots in an argument used by Lindhölm [12] in the setting of weighted Fock spaces. Let r n := 1 − 1/n, and consider a sequence of C ∞ functions χ n with compact support on D such that χ n (z) = 1 for |z| ≤ 1 − 1/n, and |∂χ n | n. For each n, consider the analytic functions where ω * is the associated weight given by Since the functions f χ n ∈ L 2 (ω) and P ω * is bounded on L p (ω p/2 ), 1 ≤ p < ∞, then f n ∈ A 2 (ω), and Therefore, it remains to show that f n → f uniformly on compact subsets of D. Since and, clearly, f χ n → f uniformly on compact subsets of D, it is enough to show that u n → 0 uniformly on compact subsets of D, with u n = f χ n − P ω * ( f χ n ). Fix 0 < R < 1 and let z ∈ D with |z| ≤ R. For n big enough, the function u n is analytic in a neighborhood of the disk D(δ 0 τ (z)), with δ 0 ∈ (0, m τ ). Hence, by Lemma A, (4.10) Since u n is the solution of the ∂-equation ∂v = f ∂χ n with minimal L 2 (ω * ) norm, by Proposition 4.5, we have Since ∂χ n is supported on r n < |z| < 1 with |∂χ n | n, we get and this goes to zero as n → ∞ since D(δτ (z)) ⊂ D and f n → f uniformly on Theorem C Suppose that ω, ω 0 , and ω 1 are weight functions on D. If 1 ≤ p 0 ≤ p 1 ≤ ∞ and 0 ≤ θ ≤ 1, then with equal norms, where With this and the result on bounded projections we can obtain the following result on complex interpolation of large weighted Bergman spaces.

Theorem 5.1
Let ω be a weight in the class E. If 1 ≤ p 0 ≤ p 1 ≤ ∞ and 0 ≤ θ ≤ 1, then is a consequence of the definition of complex interpolation, the fact that each A p k (ω p k /2 ) is a closed subspace of L p k (ω p k /2 ), and L p 0 (ω p 0 /2 ), L p 1 (ω p 1 /2 ) θ = L p (ω p/2 ). This last assertion follows from Theorem C.
On the other hand, if f ∈ A p (ω p/2 ) ⊂ L p (ω p/2 ), it follows from Theorem C that Thus, there exists a function F ζ (z) (z ∈ D and 0 ≤ Re ζ ≤ 1) and a positive constant C such that: Define a function G ζ by G ζ (z) = P ω F ζ (z). Due to the reproducing formula in Theorem 4.10, and the boundedness of the Bergman projection, see Theorem 4.1, we have: Since each function G ζ is analytic on D, we conclude that f belongs to [A p 0 (ω p 0 /2 ), A p 1 (ω p 1 /2 )]. This completes the proof of the theorem.

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 p/2 ) to identify the dual space of A p (ω p/2 ). As usual, if X is a normed space, we denote its dual by X * . For a given weight v, we introduce the space Lemma 6.1 Let ω ∈ E and z ∈ D. Then K z ∈ A 0 (ω 1/2 ).
In particular, since A 0 (ω 1/2 ) ⊂ A ∞ (ω 1/2 ) ⊂ A p (ω p/2 ), it follows that K z ∈ A p (ω p/2 ) for any p. Now we are ready to state and prove the corresponding duality results.
Here p denotes the conjugate exponent of p, that is, p = p/( p − 1).
From Fubini's theorem it is easy to see that P ω is self-adjoint. By Theorem 4.10, the reproducing formula f = P ω f holds for every f ∈ A p (ω p/2 ) ⊂ A 1 (ω 1/2 ). Then, one has Finally, the function g is unique. Indeed, if there is another function g ∈ A p (ω p /2 ) with ( f ) = g ( f ) = g ( f ) for f ∈ A p (ω p/2 ), then by testing the previous identity on the reproducing kernels K a for each a ∈ D, and using the reproducing formula, we obtain g(a) = g (K a ) = g (K a ) = g(a), a ∈ D.
Thus, any bounded linear functional on A p (ω p/2 ) is of the form = g for some unique g ∈ A p (ω p /2 ) and, furthermore, The proof is complete.
For the case of normal weights, the analogues of Theorems 6.3 and 6.5 were obtained by Shields and Williams in [18]. They also asked what happens with the exponential weights, a problem that is solved in the present paper.

Concluding Remarks
There is still plenty of work to do for a better understanding of the theory of large weighted Bergman spaces, and several natural problems are waiting for further study or a complete solution: atomic decomposition, coefficient multipliers, zero sets, etc. We hope that the methods developed here will be of some help in attacking these problems.