The Bergman kernel functions of certain unbounded domains.

Friedrich Haslinger*

Abstract. In this paper we compute the Bergman kernel functions of the unbounded domains Ω_{p} = {(z′, z) ∈ ℂ^{2} : ℑz > p(z′)}, where p(z′) = |z′|^{α}/α. It is also shown that these kernel functions have no zeroes in Ω_{p}. We use a method from harmonic analysis to reduce the computation of the 2-dimensional case to the problem of finding the kernel function of a weighted space of entire functions in one complex variable.

1. Introduction

Let Ω_{p} be a domain in ℂ^{n+1} of the form

Ω_{p} = {(z′, z) : z′ ∈ ℂ^{n} , z ∈ ℂ , ℑz > p(z′)}.

Such domains can be viewed as generalizations of the Siegel upper half space, where p(z′) = |z′|^{2} (see [S]). Weakly pseudoconvex domains of this kind were investigated by Bonami and Lohoué [BL], Boas, Straube and Yu [BSY], McNeal [McN1], [McN2], [McN3] and Nagel, Rosay, Stein and Wainger [NRSW1], [NRSW2]. For the case where p(z′) = |z′|^{k}, k ∈ ℕ, Greiner and Stein [GS] found an explicit expression for the Szegö kernel of Ω_{p}. If p is a subharmonic function on ℂ, which depends only on the real or only on the imaginary part of z′, then one can find analogous expressions and estimates in [N] (see also [Has1] ). In [D] and in [K] properties of the Szegö projection for such domains are studied. The asymptotic behavior of the corresponding Szegö kernel was investigated in [Han] and [Has2]. There have been several recent papers obtaining explicit formulas for the Bergman and Szegö kernel function on various weakly pseudoconvex domains ( [D’A], [BFS], [FH1], [FH2], [FH3] and [OPY]). From the explicit formulas one can find examples of bounded convex domains whose Bergman kernel functions have zeroes (see [BSF]). In this paper we will compute the Bergman kernel functions of the unbounded domains Ω_{p} = {(z′, z) ∈ ℂ^{2} : ℑz > p(z′)}, where p(z′) = |z′|^{α}/α, and we will also show that these kernel functions have no zeroes in Ω_{p}.

1980 Mathematics Subject Classification (1991 Revision). Primary 32A07, 32H10; Secondary 32A15.

Key words and phrases. Bergman kernel, Szegö kernel.

*Partially supported by a FWF–grant P11390–MAT of the Austrian Ministry of Sciences.

2. Computation of the Bergman kernel

We suppose that the weight function p : ℂ^{n} ⟶ ℝ_{+} is (pluri)subharmonic and of a growth behavior guaranteeing that the corresponding Bergman spaces H_{τ} of entire functions are nontrivial, where H_{τ} (τ > 0) consists of all entire functions ϕ : ℂ^{n} ⟶ ℂ such that

∫_{ℂ^{n}} |ϕ(z′)|^{2}e^{−4πτp(z′)} dλ(z′) < ∞.

The Bergman kernels of these spaces are denoted by K_{τ} (z′, w′). A result on parameter families of Bergman kernels of pseudoconvex domains of Diederich and Ohsawa [DO] can be adapted to our case, showing that for fixed (z′, w′) the function τ ↦ K_{τ} (z′, w′) is continuous. Then we can apply a method from [Has1] to obtain the following formulas for the Szegö kernel S of the Hardy space H^{2}(∂Ω_{p}) and the Bergman kernel B of the domain Ω_{p} : (see [Has3])

Proposition 1.

(a) If ∂Ω_{p} is identified with ℂ^{n} × ℝ, the Szegö kernel on ∂Ω_{p} × ∂Ω_{p} has the form

S((z′, t), (w′, s)) = ∫_{0}^{∞} K_{τ} (z′, w′)e^{−2τ(p(z′)+p(w′))}e^{−2πiτ(s−t)} dτ,

where z′, w′ ∈ ℂ^{n}, s, t ∈ ℝ.

(b) For (z′, z), (w′, w) ∈ Ω_{p} (z′, w′ ∈ ℂ^{n}; z, w ∈ ℂ) the Szegö kernel can be expressed in the form

S((z′, z), (w′, w)) = ∫_{0}^{∞} K_{τ} (z′, w′)e^{−2πiτ(\overline{w}−z)} dτ.

(c) The Bergman kernel of Ω_{p} is written as

B((z′, z), (w′, w)) = 4π ∫_{0}^{∞} τK_{τ} (z′, w′)e^{−2πiτ(\overline{w}−z)} dτ.

We first compute the Bergman kernel K_{τ} (z′, w′) of the weighted spaces of entire functions H_{τ} . Here we only consider the one-dimensional case. There are several possibilities to generalize to the higher dimensional case, where the corresponding formulas become quite complicated.

We suppose that the weight function p has the property that the Taylor series of an entire function in H_{τ} is convergent in H_{τ} . For instance, these assumptions are satisfied in the following case:

Proposition 2. Suppose that p is a convex function on ℝ^{2} = ℂ such that H_{τ} contains the polynomials, then the polynomials are dense in H_{τ} .

(see [T]) We further suppose that p depends only on |z| and has a continuously differentiable inverse ρ as a function from ℝ_{+} to ℝ_{+}. Then the Bergman kernel of H_{τ} can be computed as follows:

Proposition 3.

K_{τ} (z′, w′) = \frac{1}{2πτ} ∑_{n=0}^{∞} \frac{n + 1}{a_{n}(τ)} z^{′n} \overline{w}^{′n},

where a_{n}(τ) = L(ρ^{2n+2})(4πτ) is the Laplace transform of ρ^{2n+2} in the point (4πτ) :

L(ρ^{2n+2})(4πτ) = ∫_{0}^{∞}(ρ(s))^{2n+2} e^{−4πτs} ds.

Proof. Since the monomials (z^{′n})_{n≥0} constitute a complete orthogonal system in H_{τ} the Bergman kernel can be expressed in the form

K_{τ} (z′, w′) = ∑_{n=0}^{∞} \frac{z^{′n} \overline{w}^{′n}}{c_{n}(τ)} ,

where

c_{n}(τ) = ∫_{ℂ} |z′|^{2n} exp(−4πτp(z′)) dλ(z′)

(see [Kr] or [R]). Using polar coordinates we get

c_{n}(τ) = 2π ∫_{0}^{∞} r^{2n+1} exp(−4πτp(r)) dr,

after substituting p(r) = s we obtain

c_{n}(τ) = 2π ∫_{0}^{∞}(ρ(s))^{2n+1} exp(−4πτs) ρ′(s) ds.

Now partial integration gives

2π ∫_{0}^{∞}(ρ(s))^{2n+1} exp(−4πτs) ρ′(s) ds = \frac{2πτ}{n + 1} ∫_{0}^{∞}(ρ(s))^{2n+2} exp(−4πτs) ds,

which proves the proposition. □

In the next step we compute the Bergman kernel of Ω_{p} ⊂ ℂ^{2} :

Proposition 4. Let the weight function p be as in Proposition 3. Then the Bergman kernel B((z′, z), (w′, w)) of Ω_{p} = {(z′, z) ∈ ℂ^{2} : ℑz > p(z′)} can be written in the form

B((z′, z), (w′, w)) = 2 ∫_{0}^{∞} (∑_{n=0}^{∞}(n + 1)\frac{e^{−2πi(\overline{w}−z)τ}}{L(ρ^{2n+2})(4πτ)} z^{′n} \overline{w}^{′n}) dτ.

Proof. Combine Proposition 1 (c) and Proposition 3. □

In the sequel we concentrate on weight functions of the form p(z′) = |z′|^{α}/α, where α ∈ ℝ , α ≥ 1. It is easily seen that in this case the assumptions of Proposition 2 and Proposition 3 are satisfied. Hence we can apply Proposition 4 and get

Proposition 5. Let p(z′) = |z′|^{α}/α, where α ∈ ℝ , α ≥ 1. Then the Bergman kernel B((z′, z), (w′, w)) of Ω_{p} = {(z′, z) ∈ ℂ^{2} : ℑz > p(z′)} has the form

B((z′, z), (w′, w))

= \frac{2}{π(i(\overline{w} − z))^{2}} \begin{matrix}[\frac{αi}{2}(\overline{w} − z)] & 2/α & [(2 + α) [\frac{αi}{2}(\overline{w} − z)]^{2/α} + (2 − α) z′ \overline{w}′] \\ [[\frac{αi}{2}(\overline{w} − z)]^{2/α} − z′ \overline{w}′] & 3 \end{matrix}.

We always take the principal values of the multi-valued functions involved.

Proof. First we compute the Laplace transform L(ρ^{2n+2})(4πτ). In our case we have ρ(s) = (αs)^{1/α}, hence

L(ρ^{2n+2})(4πτ) = ∫_{0}^{∞}(αs)^{(2n+2)/α} e^{−4πτs} ds
= (4πτ)^{−1−(2n+2)/α} α^{(2n+2)/α} ∫_{0}^{∞} t^{(2n+2)/α} e^{−t} dt
= (4πτ)^{−1−(2n+2)/α} α^{(2n+2)/α} Γ(1 + (2n + 2)/α).

In the sequel of the proof it will become apparent that summation and integration in Proposition 4 can be interchanged. We obtain now

B((z′, z), (w′, w))

= 2 ∑_{n=0}^{∞} \frac{(n + 1) (4π)^{1+(2n+2)/α}}{α^{(2n+2)/α} Γ(1 + (2n + 2)/α)} (∫_{0}^{∞} τ^{1+(2n+2)/α} e^{−2πi(\overline{w}−z)τ} dτ) z^{′n} \overline{w}^{′n}.

The integral in the brackets can be expressed in the form

∫_{0}^{∞} τ^{1+(2n+2)/α} e^{−2πi(\overline{w}−z)τ} dτ
= (2πi(\overline{w} − z))^{−2−(2n+2)/α} ∫_{0}^{∞} σ^{1+(2n+2)/α} e^{−σ} dσ,

since ℜ(2πi(\overline{w} − z)) > 0, this follows by Cauchy’s theorem (see for instance [He], pg.33). Now we obtain

∫_{0}^{∞} τ^{1+(2n+2)/α} e^{−2πi(\overline{w}−z)τ} dτ
= (2πi(\overline{w} − z))^{−2−(2n+2)/α} Γ(2 + (2n + 2)/α)

= (2πi(\overline{w} − z))^{−2−(2n+2)/α} (1 + (2n + 2)/α) Γ(1 + (2n + 2)/α).

We can now continue computing the Bergman kernel

B((z′, z), (w′, w))

= 2 ∑_{n=0}^{∞} \frac{(n + 1) (1 + (2n + 2)/α) (4π)^{1+(2n+2)/α}}{α^{(2n+2)/α} (2πi(\overline{w} − z))^{2+(2n+2)/α}} z^{′n} \overline{w}^{′n}

= \frac{2}{π} ∑_{n=0}^{∞} \frac{2^{(2n+2)/α} [2(n + 1)^{2}/α + (n + 1)]}{α^{(2n+2)/α} (i(\overline{w} − z))^{2+(2n+2)/α}} z^{′n} \overline{w}^{′n}

= \frac{2}{π(i(\overline{w} − z))^{2}} ∑_{n=0}^{∞} [\frac{2(n + 1)^{2}}{α} + (n + 1)] [\frac{αi}{2}(\overline{w} − z)]^{−2(n+1)/α} z^{′n} \overline{w}^{′n}.

For the summation we use the formulas

∑_{n=0}^{∞}(n + 1)^{2} x^{n} = \frac{1 + x}{(1 − x)^{3}}  and  ∑_{n=0}^{∞}(n + 1) x^{n} = \frac{1}{(1 − x)^{2}} ,

where |x| < 1. Sine ℑz > |z′|^{α}/α and ℑw > |w′|^{α}/α it follows that

|z′w′| < |\frac{αi}{2}(\overline{w} − z)|^{2/α}

and hence

B((z′, z), (w′, w))

= \frac{2}{π(i(\overline{w} − z))^{2}} \begin{matrix}[\frac{αi}{2}(\overline{w} − z)] & −2/α & [2 + α + (2 − α) [\frac{αi}{2}(\overline{w} − z)]^{−2/α} z′ \overline{w}′] \\ [1 − [\frac{αi}{2}(\overline{w} − z)]^{−2/α} z′ \overline{w}′] & 3 \end{matrix}

= \frac{2}{π(i(\overline{w} − z))^{2}} \begin{matrix}[\frac{αi}{2}(\overline{w} − z)] & 2/α & [(2 + α) [\frac{αi}{2}(\overline{w} − z)]^{2/α} + (2 − α) z′ \overline{w}′] \\ [[\frac{αi}{2}(\overline{w} − z)]^{2/α} − z′ \overline{w}′] & 3 \end{matrix},

which proves Proposition 5. □

Proposition 6. Let p(z′) = |z′|^{α}/α, where α ∈ ℝ , α ≥ 1. Then the Bergman kernel B((z′, z), (w′, w)) of Ω_{p} = {(z′, z) ∈ ℂ^{2} : ℑz > p(z′)} has no zeroes in Ω_{p}.

Proof. By Proposition 5 the Bergman kernel B((z′, z), (w′, w)) has a zero if and only if

[\frac{αi}{2}(\overline{w} − z)]^{2/α} = \frac{α − 2}{α + 2} z′ \overline{w}′.

Since ℑz > 0 and ℑw > 0, the factor (\overline{w} − z) never vanishes on Ω_{p}. So we have a contradiction in the case α = 2. Now suppose that α ≠ 2. If the Bergman kernel has a zero, then

|\frac{αi}{2}(\overline{w} − z)|^{2} = |\frac{α − 2}{α + 2}|^{α} |z′|^{α} |\overline{w}′|^{α}.

We set w = u + iv , z = x + iy and know that αy > |z′|^{α} and αv > |w′|^{α}, hence

(u − x)^{2} + (v + y)^{2} < 4 |\frac{α − 2}{α + 2}|^{α} vy.

Since both v and y are larger than 0 and since 4vy ≤ (v + y)^{2}, this inequality can only hold if at least

1 < |\frac{α − 2}{α + 2}|^{α} .

It is clear that the last inequality is false, so the Bergman kernel has no zeroes in Ω_{p}. □

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Institut für Mathematik , Universität Wien, Strudlhofgasse 4, A-1090 Wien, Austria E-mail: HAS@PAP.UNIVIE.AC.AT, http://radon.mat.univie.ac.at/˜ fhasling