Pointwise estimates for the Bergman kernel of the weighted Fock space

We prove upper pointwise estimates for the Bergman kernel of the weighted Fock space of entire functions in $L^2(e^{-2\phi})$ where $\phi$ is a subharmonic function with $\Delta \phi$ a doubling measure. We derive estimates for the canonical solution operator to the inhomogeneous Cauchy-Riemann equation and we characterize the compactness of this operator in terms of $\Delta \phi$.


INTRODUCTION
Let φ be a subharmonic function in C whose Laplacian ∆φ is a doubling measure. For 1 ≤ p < ∞, we consider the Fock spaces where dm denotes the Lebesgue measure in C.
In this context the Bergman kernel has already been studied. In [Chr91] M. Christ obtained pointwise estimates under the hipothesis that φ is a subharmonic function such that µ = ∆φ is a doubling measure and (1) inf z∈C µ(B(z, 1)) > 0.
This result was extended to several complex variables by H. Delin and N. Lindholm in [Del98] and [Lin01] under similar hypothesis. They obtain a very fast decay of the Bergman kernel away from the diagonal.
We will remove hypothesis (1) (which in somes sense is related to the strict pseudoconvexity) and keep only the doubling condition (that is morally closer to finite-type). We still obtain some decay away from the diagonal, we derive estimates for the canonical solution operator to the inhomogeneous Cauchy-Riemann equation and we characterize the compactness of this operator in terms of ∆φ. Our main result is the following estimate.
Although the estimate above seems to be asymmetric in the variables z, ζ one can see that for |z − ζ| < C max{ρ(z), ρ(ζ)} the values of ρ(z) and ρ(ζ) are comparable, see Lemma 2.3. Also when |z − ζ| ≥ C max{ρ(z), ρ(ζ)} one can use Lemma 2.6 to see that the same estimate holds with ρ(ζ) inside the exponential for a different positive exponent ǫ (this new exponent depending only on the doubling constant for ∆φ). The symmetry becomes apparent when we write (2) in terms of the distance d φ induced by the metric ρ −2 φ (z)dz ⊗ dz. Indeed, by using Lemma 2.6 one can write (2) as for some ǫ > 0 (different from the previous one but still positive). The estimate proved in [Chr91] for the Bergman kernel of F 2 φ defined for a φ with doubling Laplacian and satisfying (1) is for some ǫ > 0 and all z, ζ ∈ C. Let N be the canonical solution operator to ∂, i.e. ∂Nf = f and Nf is of minimal L 2 (e −2φ ) norm and let C(z, ζ) be the integral kernel such that The boundedness and compactness of this canonical solution operator from L 2 (e −2φ ) to itself has been extensively studied in one and several variables; for a survey on this problem and its applications see [FS02]. It is shown in [Has06] that for weights on the class considered by M. Christ, the condition ρ(z) → 0 when |z| → ∞ is sufficient for compactness. In the same paper it is shown that the canonical solution operator with φ(z) = |z| 2 fails to be compact, all these results are contained in Theorem 1.3. Finally, in [HH07] the authors prove a result similar to Theorem 1.3 with some extra regularity conditions on ∆φ.
With Theorem 1.1 we obtain a pointwise estimate on the kernel of the canonical solution operator.
Theorem 1.2. There exists an integral kernel G(z, ζ) such that solves ∂u = f and Moreover, the integral kernel C(z, ζ) giving the canonical solution to ∂ in L 2 (e −2φ ) has the same estimate (with a different exponent ǫ > 0).
One can compare this result with the estimate on [Chr91, Theorem 1.13] where the author proves that As an application of the estimate (2) we characterize the compactness of the canonical solution operator to ∂ in terms of the measure ∆φ. Theorem 1.3. Let φ be a subharmonic function such that ∆φ is doubling. The canonical solution operator N of minimal norm in L 2 (e −2φ ) to the inhomogeneous ∂-equation defines a bounded compact operator from L 2 (e −2φ ) to itself if and only if ρ φ (z) → 0 when |z| → ∞.
Any of the estimates on C(z, ζ) (the estimate in Theorem 1.2 or the result by Christ, (4)) can be used in order to prove this theorem, because as soon as one supposes the compactness of the canonical solution operator N, the function ρ turns out to be bounded and therefore (1) holds.
There is some natural gain (or loss) in the Hörmander estimates if the Laplacian of φ is big (or small). If we incorporate the Laplacian in the weight then we always get boundedness, under some mild regularity assumption (the doubling property) but we never get compactness: f e −φ ρ L p (C) , for all p ∈ [1, ∞]. Moreover, the solution operator N acting from L 2 (e −2φ ρ 2 ) to L 2 (e −2φ ) is always bounded but it is never compact.
Remark. The first statement in this proposition has been proved already in [MMO03, Theorem C] by using peak functions instead of estimates for the Bergman kernel.

PRELIMINAIRES
In this section we collect some material from [Chr91] and [MMO03] that will be used along the proofs and we deduce some easy estimates for the Bergman kernel near the diagonal. for all z ∈ C and r > 0. The smallest constant C in the previous inequality is called the doubling constant for µ.
Lemma 2.2. [Chr91, Lemma 2.1] Let µ be a doubling measure in C. There exists a constant γ > 0 such that for any disks D, D ′ with respective radius r > r ′ and with D ∩

Remark.
In particular for any z ∈ C and r > 1 there exists a constant γ > 0 (depending only on the doubling constant for µ) such that (5) r γ µ(D r (z)) r 1/γ .
It follows inmediately from Lemma 2.2 that the function ρ is nearly constant on balls.
, with constants depending only on the doubling constant for ∆φ.
Remark. There exist constants η, C > 0 and 0 < β < 1 such that The following lemma shows that our main estimate (2) is symmetric in the variables z, ζ.
for some 0 < δ < 1 depending only on the doubling constant for ∆φ.
Lemma 2.6. There exists δ > 0 such that for every r > 0 there exists C r > 0 such that The following lemma will be used repeatedly in what follows.
Lemma 2.7. Let φ be a subharmonic function with µ = ∆φ doubling. Then for any ǫ > 0 and where C > 0 is a constant depending only on k, ǫ, and on the doubling constant for µ.
We will also use some Cauchy-type estimates for functions in the space, The following result proved in [MMO03,Theorem 14] shows that the same space F 2 φ can be defined with a more regular weight. Proposition 2.9. Let φ be a subharmonic function such that ∆φ is doubling. There exists φ ∈ C ∞ (C) such that |φ − φ| ≤ C with ∆ φ doubling and As a first step in proving Theorem 1.1, in the remainder of the section we derive some estimates for the Bergman kernel on the diagonal or near the diagonal.
Proposition 2.10. There exist C > 0 such that Proof. Let z ∈ C be fixed. For any M ∈ N there exists a holomorphic function P z such that P z (z) = 1 and [MMO03,Appendix]. For some c 0 > 0 (to be determined) we define the entire function The other estimate follows by using the reproducing property for the Bergman kernel, Lemma 2.3 and inequality (a) in Lemma 2.8, see the next proposition, where this is done in detail.
The following coarse estimate will give us (2) when the points z, ζ ∈ C are close to each other.
Proposition 2.11. Let K(z, ζ) be the Bergman kernel for F 2 φ . Then there exists C > 0 (depending only on the doubling constant for ∆φ) such that for any z, ζ ∈ C Proof. Let z ∈ C be fixed. Applying (a) in Lemma 2.8 to the reproducing kernel K z and using Lemma 2.3 Finally, by using Proposition 2.10 the estimate follows.
3. PROOF OF THEOREM 1.1 We will follow a similar argument as in [Lin01] when Lindholm studies the case when ∆φ is bounded. In fact the basic trick goes back to Kerzman in [Ker72], where the Bergman kernel is estimated using the estimates on the solution to an inhomogeneous Cauchy-Riemann equation.
We are interested in studying the behaviour of K(z, ζ) when the points z, ζ are far apart. .
We have that where P = P φ stands for the Bergman projection and, since χ(z) = 0, one has We use a Hörmander's type theorem to majorize this last expression by an integral involving f ∂χ. One technical difficulty is that our function φ is not smooth enough, so first of all we define a regularized version. Let 0 < ǫ < 1 a constant to be chosen later. Let (we will write ϕ ǫ if we need to stress the dependence on ǫ). The function ϕ is subharmonic and .
Considering the dependence on ǫ one has The Laplacian of ϕ is not bounded above, so we define We denote Φ = ∆ψ 2ǫ (w)/4 and, as before, we will write Φ ǫ if needed. By [MMO03, Theorem 14] one can build φ ∈ C ∞ such that |φ − φ| ≤ C with ∆ φ doubling and Lemma 3.1. There exist 0 < ǫ 0 < 1 and 0 < C 1 , C 2 < 1 (depending only on the doubling constant for ∆φ) such that for any 0 < ǫ ≤ ǫ 0 This Lemma is an easy consequence of the following: Lemma 3.2. For any C > 0 there exists 0 < ǫ 0 < 1 (depending only on the doubling constant for ∆φ and C) such that Lema 3.2 implies Lemma 3.1. By (9) let C ′ > 0 such that Let ǫ 0 > 0 the one provided by Lemma 3.2 for C > 0 such that Then it is enough to take C 1 = CC ′ and C 2 = 4CC ′ .
or equivalently and this follows from Lemma 2.4 because ζ ∈ D(w). We would like to mention that, as |w−ζ| > ρ(ζ), the last inequality holds also for any exponent smaller than δ. Finally, as 2 ǫ C/ǫ goes to infinity when ǫ → 0, one can find ǫ 0 such that (10) holds for any 0 < ǫ ≤ ǫ 0 .
From now on we will fix ε > 0 in such a way that the conclusions of Lemma 3.1 do hold. The following Lemma is an easy consequence of the previous ones.
As D φ (ζ) ∩ D φ (z) = ∅, the function f χ vanishes off D φ (ζ) and therefore (recall that by Lemma 3.3 ρ ̺ ∼ ρ e φ ∼ ρ φ ) the function u is holomorphic in D r ̺ (z) for some r > 0 again by (a) in Lemma 2.8 We estimate this last integral using the classical Hörmander theorem:

Theorem 3.4 (Hörmander).
Let Ω ⊂ C be a domain and φ ∈ C 2 (Ω) be such that ∆φ ≥ 0. For any f ∈ L 2 loc (Ω) there exist a solution u to ∂u = f such that and also with a variant due to Berndtsson (see [Ber01, Lemma 2.2]): and for any g one can find v such that ∂v = g with then (for v 0 ) the canonical solution in L 2 (e −2φ ), one has where C = 6/(1 − C 1 ) 2 .
We know that ∆( φ + ψ) ≥ 0,then applying Theorem 3.4, to ∂(f χ),one has v such that ∂v = ∂(f χ) with As |φ − φ| ≤ C we have that (12) holds and by Theorem 3.5 The functions φ, φ are pointwise equivalent and ∆ φ ∼ ρ −2 φ so one can estimate (11) as The function ψ is bounded above in D(ζ) by a constant depending only on the doubling constant for ∆φ, indeed, for w ∈ D(ζ) So finally (13) can be estimated by . and we have 3.1. Pointwise estimates. In this subsection we deduce a new expression, without ψ, for (14). The new expression is the one appearing in Theorem 1.1 and therefore this will finish the proof.

PROOF OF THEOREMS 1.2 AND 1.3
Theorem 1.2. Let {z i } be a sequence of points in C and r > 0 such that {D r (z j )} is a covering of C. Let {χ i } be a partition of unity subordinate to the covering. Let k z (ζ) = K(z, ζ)/ K(z, z) be the normalized reproducing kernel in F 2 φ . Consider the operator By Cauchy-Pompeiu formula one has that ∂u i = f χ i . Then the kernel is such that solves ∂u = f . Let z ∈ C be fixed and |z − ζ| ≤ Rρ(z) for some fixed R ≫ 0, then there is a finite number of balls of the covering intersecting D(z) and by Proposition 2.11 one has |G(z, ζ)| |z − ζ| −1 .
Also when |z − ζ| ≥ Rρ(z) there is a finite number of balls in the covering containing ζ and this will give us a finite number of summands in G. For one of these terms one has by Theorem 1.1 that ζ) this gives us the estimate of G.
Now we want to show that the same estimate holds for the kernel C. If N is the canonical solution operator and M is the solution operator given by the kernel G above, one can see that N = M − P M where P stands for the Bergman projection. Then for Suppose first that |z − ζ| ≤ ρ(z). We split the last integral and use the estimates on G and the Bergman kernel and we get that the first integral is bounded by a constant, where K > 1 is such that D(z) ⊂ D K (ζ). Now by Proposition 2.4 there exists ǫ ′ > 0 such that where for the first integral we use that {ξ : |ζ − ξ| < ρ(ξ)} ⊂ D K ′ (ζ) for some K ′ > 0 and for the second one we use Lemma 2.7 together with Proposition 2.9 getting For |z − ζ| > ρ(z) and given 0 < η < 1 we split the integral in the regions defined by and the last integral can be bounded as above. An entirely analogous argument proves case (ii). Let A be denote the region defined by (iii) (in the estimates which follow the value of the exponent ǫ may change from line to line although it is always strictly positive) Theorem 1.3. Let {z j } be a sequence of complex numbers such that z j → ∞ for j → ∞. We want to show that ρ(z j ) → 0 when N is compact. By Theorem 1.1 Defining holomorphic (0, 1)−forms f j and functions u j as then ∂u j = f j . Observe that u j ∈ F 2 φ because of the above estimate, Lemma 2.7 and Proposition 2.9 C |z − z j | 2 |k z j (z)| 2 e −2φ(z) dm(z) ρ 2 (z j ) < ∞.
Finally, as the reproducing kernels {k w } w∈C are dense in F 2 φ and the solution u j is the canonical solution to ∂ i.e. u j = Nf j . By hypothesis, the operator N is compact and f j = 1 and therefore there exist a convergent subsequence of {u j } (which we denoted as before).
The functions u j are basically concentrated on D(z j ). Indeed, by Proposition 2.11 one has and conversely by Lemma 2.8 (c) In particular, just because the operator N is bounded, the sequence {ρ(z j )} has to be bounded. Also by Lemma 2.7 and Proposition 2.9 one has where C r → 0 when r → ∞. The sequence {u j } is a Cauchy sequence so for j, k → ∞. To complete this part of the proof we have to see that the scalar product is small also when z j and z k are far enough from each other. Indeed, given ǫ > 0 there exists r ǫ such that for r ≥ r ǫ Now let |z j −z k | ≫ r ǫ max{ρ(z j ), ρ(z k )}. The L 2 −norm of u j on D r (z j ) is pointwise equivalent to ρ(z j ) (and this value is bounded above) so applying Hölder's inequality to we deduce that the scalar product is arbitrarily small and Suppose now that ρ(z) → 0 when |z| → ∞ and let be such a solution operator, i.e. ∂Mf = f . If M is compact then the canonical solution operator will be compact because it can be written as N = M − P M where P is the Bergman projection. So all we have to show is that there exists a solution operator for the ∂ problem which is compact. First of all, the operator M δ : L 2 (e −2φ ) → L 2 (e −2φ ) defined as has norm O(δ) as δ → 0. Indeed, let z ∈ C be fixed, then where the constant C only depends on the doubling constant for ∆φ. Also C |M δ f (z)|e −φ(z) dm(z) δ ρ ∞ f e −φ L 1 (C) , and by Marcinkiewicz interpolation theorem, when ρ is bounded, the norm of the operator from L 2 (e −2φ ) to L 2 (e −2φ ) is O(δ).
For the L 1 norm Assume now that the operator is compact. Let {z j } be a sequence of complex numbers such that the disks D(z j ) are pairwise disjoint. If f j (z) = k z j (z) ρ(z j ) dz, u j (z) = (z − z j ) k z j (z) ρ(z j ) , one has ∂u j = f j and C |f j (z)| 2 e −2φ(z) ρ(z) dm(z) 1 and one can extract a converging subsequence of {u j }. But as before, from u j − u k 2 = u j 2 + u k 2 + 2 Re u j , u k → 0 we get a contradiccion because u j ∼ 1 and | u j , u k | → 0 for a fixed k when j → ∞.