The Weighted Bergman Kernel and the Green’s Function

We study the connection between the weighted Bergman kernel and the Green’s function on a domain W⊂C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W\subset \mathbb {C}$$\end{document} for which the Green’s function exists.

It is known, in the classical case, that (see [19]) for z, w ∈ W, z = w (it was originally proved in [2] with additional assumptions on ∂ W ). On the other hand, if ∂ W consists of a finite number of Jordan curves, ρ(z) is a positive continuously differentiable function of x and y on a neighborhood of W , K W, ρ (z, w) a weighted Bergman kernel of the space L 2 H (W, ρ) and G W, ρ the Green's function for an operator P ρ = ∂ ∂z , then (see [6]). A very natural question is: "can we relax the regularity of ∂ W ?". In this paper we prove that the connection above holds for any domain W ⊂ C, for which L 2 H (W ) = 0 (any bounded domain share this property) and for weights ρ such that log ρ is harmonic. The generalization is not straightforward, since we need to find the connection between a classical Green function and G W, ρ . We shall begin with the definitions and basic facts used in this paper. Additionally, because we are dealing with the weighted Bergman kernels, we will recall for which weights in general the weighted Bergman kernel exists (although we are working here with differentiable weights only).

Definitions and Notation
Let W ⊂ C be a domain, and let W(W ) be the set of weights on W , i.e., W(W ) is the set of all Lebesque measurable, real-valued, positive functions μ on W (we consider two weights as equivalent if they are equal almost everywhere with respect to the Lebesque measure on W ). For μ ∈ W(W ) we denote by L 2 (W, μ) the space of all Lebesque measurable, complex-valued, μ-square integrable functions on W , equipped with the norm || · || W,μ := || · || μ and given by the scalar product Let us recall the definition (Definition 2.1) of admissible weight given in [13].
is called an admissible weight, an a-weight for short, if L 2 H (W, μ) is a closed subspace of L 2 (W, μ) and for any z ∈ W , the evaluation functional E z is continuous on L 2 H (W, μ). The set of all a-weights on W will be denoted by AW(W ).
The definition of admissible weight provides us with existence and uniqueness of the related Bergman kernel and completeness of the space L 2 H (W, μ). The concept of a-weight was introduced in [12], and in [13] several theorems concerning admissible weights are proved. An illustrative result is: It can be proved, in a fashion similar to the classical case, that if μ is an admissible weight then there exists exactly one function minimizing the norm. Let us denote it by φ μ (z, t). The weighted Bergman kernel function K W, μ is defined as follows:

From the Unweighted to the Weighted Case
Let us recall that we are working with a domain W ⊂ C, for which L 2 H (W ) = 0 (any bounded domain has this property). We define the Green's function G W, ρ as the limit in C 2 (W ) of the sequence {G W j , ρ j } ∞ j=0 , for an arbitrary exhaustion {W j } of W by domains with the boundary consisting of a finite number of smooth Jordan curves, and for ρ j = ρ |W j . We are assuming here that the limit exists and is independent of the exhaustion. Now we will use the result from [19] to prove the following , and has no zeros on W , then Proof It is well known that any domain W ⊂ C may be written as where ∂ W j consists of a finite number of smooth Jordan curves (we do not assume any regularity of ∂ W ), for any j ∈ N. Let ρ j (z) = |μ j (z)| 2 where μ j = μ |W j . One can find in ( [6], p. 494) that . Moreover (by a standard calculation) Let us note that L 2 because the image of a reproducing kernel by a unitary map is the corresponding reproducing kernel in the target space (this may be also seen by using the com- ρ (z, w)). Now Theorem 3.1 follows from the result in [19].

Non-Holomorphic Weights
On closer scrutiny, the crucial thing in the proof of Theorem 3.1 was to relate the weighted Green's function to the unweighted one. However, that was possible since holomorphicity of μ allowed us to find a "bridge" between Green's functions. This relationship turns out to be preserved even if we relax the assumption about holomorphicity of the weight. We will do some reduction which transforms the problem to solving some PDE. It turns out to be possible if only log ρ is harmonic, as the following reveals.

Theorem 3.2 If ρ(z) = |μ(z)| 2 , where log ρ is harmonic on a neighborhood of W (and μ has no zeros on W ) then
Proof Let {W j } ∞ j=1 be an exhaustion of W and ρ j = |μ j | 2 where μ j = μ |W j . The crucial thing is to find g j (z) such that u j (w) = g j (w)U j (w) is a general solution of the equation and U j (w) is (an arbitrary) complex and harmonic function on W j (we define g on the same way by means of P ρ ). Thus Remark 3.3 By the equation above, g j is an antiholomorphic function.
Examining the system above, we see that the first equation is a consequence of the second one. Let us focus on the third one: It may be written in the form Thus, for a given μ j , there is a function g j which must satisfy: Notice that, if μ j is holomorphic and g j = μ j , then the system above is satisfied (in this case we get the result of [6]). We may proceed to get the exact form of g j (z), namely: where h j ∈ O(W j ). Since the g j need to be antiholomorphic, h j is not an arbitrary holomorphic function. Let us proceed to get the exact form of h j .
where l j is antiholomorphic. Taking ∂ 2 ∂w∂w we see that log |μ j | must be harmonic.
Write log |μ j (w)|dy = F j (x, y) + c(x), where w = x + iy and ∂ ∂ y F j (x, y) = log |μ j (w)| 2 . Since log |μ j | 2 is harmonic, we have where d j (x)dx = e j (x) + c(y). We may easily check that Im h j (w) is harmonic.
To this end, where ∂ ∂ y F j (x, y) = log |μ j (w)| 2 , and e j (x) = d j (x). Thus where h j is given above. Now, by the definition of μ j we have that and by Harnack's theorem log |μ| is harmonic. Again (as in [6]) By the regularity of any ∂ W j we have which in the limit as j → ∞ yields ∂z∂w .
(we made use of the result in [20] for the LHS of the above).

Remarks and Some Applications
One could try the "reduction" used in the proof above to find a connection between the classical Green's function and the Green's function of some other differential operator of elliptic type. It is well established that weighted Bergman spaces are both intrinsically interesting and a powerful analytic tool. Our purpose in this paper has been to develop this set of ideas, and particularly the connection between the Bergman kernel and the Green's function in the weighted context. Some of the applications might be: (a) With the established connection between weighted Bergman kernel and Green's function in hand, we can reformulate the weighted version of the so called "small conjecture" (Is the so called Skwarczyński distance equivalent to the Bergman distance?-see [16,17]) as: Remark 4.1 Assume W C, and μ is a continuously differentiable function of x and y on a neighborhood of W . Then t n → t ∈ ∂ W represents defective evaluation (see [17]) iff − 2 πρ(z)ρ(w) ∂ 2 ∂z∂w G W, ρ (z, w)(·, t n ) → γ weakly in L 2 H (W, μ) and where ||γ || = κ. This is important, since the involved so-called Skwarczyński distance is biholomorphically invariant, and given more explicitly than the Bergman distance.
(b) Using the method of alternating projections (see [15]), we can recover (having some Dirichlet and Neuman boundary conditions on G W ) G W,μ for an arbitrary domain W lying in C.