Bergman and Szegö kernels for certain unbounded domains in ℂ^{2}.

Friedrich Haslinger*

In this expository article we consider unbounded weakly pseudoconvex domains in ℂ^{2} of the form

Ω_{p} = {(z_{1}, z_{2}) ∈ ℂ^{2} : ℑ(z_{2}) > p(z_{1})},

where p : ℂ ⟶ ℝ_{+} is a suitable C^{1}–function. Weakly pseudoconvex domains of this kind were investigated by McNeal [McN1] and Nagel, Rosay, Stein and Wainger [NR-SW1],[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, 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], [K] and in [Z] properties of the Szegö projection for such domains are studied. The asymptotic behavior of the corresponding Szegö kernel was investigated in [Han], [Has2] and [Kam].

1. Hardy spaces and extension of CR-distributions.

Let H^{2}(∂Ω_{p}) denote the subspace of L^{2}(∂Ω_{p}) consisting of boundary values of holomorphic functions f on Ω_{p} such that

sup_{y>0} ∫_{ℂ} ∫_{ℝ} |f(z, t + ip(z) + iy)|^{2} dλ(z)dt < ∞,

where dλ denotes the Lebesgue measure on ℂ. We identify ∂Ω_{p} with ℂ × ℝ and note that for each f ∈ H^{2}(∂Ω_{p}) there exists a boundary function f_{0} on ∂Ω_{p} such that f_{y}(z, t) := f(z, t + ip(z) + iy) tends to f_{0}(z, t) in L^{2}(ℂ × ℝ) as y tends to 0, moreover we have

∫_{ℂ} ∫_{ℝ} |f_{0}(z, t)|^{2} dtdλ(z) = sup_{y>0} ∫_{ℂ} ∫_{ℝ} |f(z, t + ip(z) + iy)|^{2} dtdλ(z)

1980 Mathematics Subject Classification (1991 Revision). Primary 32A07, 32A35, 32H10; Secondary 30D20, 32A15, 42B10.

Key words and phrases. Model domains, CR functions, tangential Cauchy-Riemann equation, relative fundamental solution, Bergman kernel, Szegö kernel, Hardy spaces, Hilbert spaces of entire functions.

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

(see [M], [SW]).

We consider the tangential Cauchy–Riemann operator for ∂Ω_{p}

\overline{L} = \frac{∂}{∂\overline{z_{1}}} − 2i\frac{∂p}{∂\overline{z_{1}}} (z_{1})\frac{∂}{∂\overline{z_{2}}} .

After the identification of ∂Ω_{p} with ℂ × ℝ the tangential Cauchy–Riemann operator has the form

\overline{L} = \frac{∂}{∂\overline{z}} − i\frac{∂p}{∂\overline{z}} \frac{∂}{∂t} .

Distributions ϕ satisfying \overline{L}(ϕ) = 0 are called CR-distributions (see [B]). The following result shows that each function f ∈ L^{2}(∂Ω_{p}) which is also a CR-distribution can be extended to a function holomorphic on Ω_{p} which belongs to H^{2}(∂Ω_{p}). The extension is expressed in terms of a corresponding entire function with a growth condition depending on p. Let E_{p} denote the space measurable functions F on ℂ × ℝ_{+} being entire with respect to the first variable and satisfying

∫_{0}^{∞} ∫_{ℂ} |F (z, t)|^{2} exp(−4πtp(z)) dλ(z)dt < ∞.

Here and in what follows we have to make sure that the weight function p is chosen in a way that the corresponding space of entire functions E_{p} is nontrivial, for instance if p grows like |z|^{α} (α > 0) for |z| → ∞. The following theorem is a version of an important representation result for Hardy spaces (see [SW] for a special case).

Theorem 1. Every function in H^{2}(∂Ω_{p}) has the representation

f(z, w) = ∫_{0}^{∞} F (z, t)e^{2πitw} dt ,

where F ∈ E_{p}. In addition F can be gained back with the help of the boundary value of f by

F (z, τ) = ∫_{ℝ} f(z, t + ip(z))e^{−2πiτt}e^{2πτp(z)} dt .

For a function g ∈ L^{2}(dλ(z)dt) let F denote the Fourier transform with respect to the variable t ∈ ℝ :

(Fg)(z, τ) = ∫_{ℝ} g(z, t)e^{−2πitτ} dt .

Then

F\overline{L}F^{−1} = \frac{∂}{∂\overline{z}} + τ \frac{∂p}{∂\overline{z}} .

F and F^{−1} are to be taken in the sense of the Plancherel theorem. Now let M denote the multiplication operator

M : L^{2}(dλ(z)dt) ⟶ L^{2}(e^{−4πtp(z)}dλ(z)dt)

defined by

(Mg)(z, τ) = e^{2πτp(z)}g(z, τ) ,

for g ∈ L^{2}(dλ(z)dt) . The proof of Theorem 1 can be derived from the fact that

(1)  F\overline{L}F^{−1} = M^{−1} \frac{∂}{∂\overline{z}} M .

Theorem 2. Let f ∈ L^{2}(∂Ω_{p}) and suppose that \overline{L}(f) = 0 in the sense of distributions. Then f can be extended to a function holomorphic in Ω_{p}, in fact belonging to H^{2}(∂Ω_{p}), whose boundary value coincides with the original function.

For the proof we use Theorem 1 and we define a function F on ℂ × ℝ_{+} by

F (z, τ) = ∫_{ℝ} f(z, t + ip(z))e^{−2πiτt}e^{2πτp(z)} dt,

and conclude from (1) that F is entire with respect to z and belongs to E_{p} and

h(z, w) = ∫_{0}^{∞} F (z, τ)e^{2πiτw} dτ,

for (z, w) ∈ Ω_{p} yields the desired extension.

Remark: If f belongs to H^{2}(∂Ω_{p}), then its boundary value f_{0} satifies \overline{L}(f_{0}) in the sense of distributions. Hence H^{2}(∂Ω_{p}) can be identified with the space of all functions f ∈ L^{2}(∂Ω_{p}) satisfying \overline{L}(f) = 0 in the sense of distributions. The Szegö projection

S : L^{2}(∂Ω_{p}) ⟶ H^{2}(∂Ω_{p})

can therefore be viewed as the projection onto the kernel of the tangential Cauchy-Riemann operator. For further details see [Has3].

2. Regularity of the relative fundamental solution operator for L.

The object of this section investigating the relative fundamental solution operator for L in two special cases, namely for p(z) = |z|^{2k} and for p(z) = x^{2k}, where z = x + iy and k ∈ ℕ. In the first case the operator L has the form

L = \frac{∂}{∂z} + ik\overline{z}|z|^{2(k−1)} \frac{∂}{∂t} .

If k = 1, then L is the well known Lewy operator on ℝ^{3} = ℂ × ℝ. See [GS] and [S] for many interesting results. Using Fourier series decompositions it can be shown that the relative fundamental solution operator can be expressed in terms of a integral operator with respect to a kernel function.

Theorem 3 ([K]). Let A = \frac{1}{2}[|z|^{2k} + |w|^{2k} − i(t − s)] and B = \frac{1}{2}[|z|^{2k} − |w|^{2k} + i(t − s)]. And let

K((z, t), (w, s)) = \frac{1}{4π^{2}B} [\frac{z}{A^{1/k} − z\overline{w}} + \frac{w}{\overline{A}^{1/k} − \overline{z}w} ] .

Define an operator T by

T f(z, t) = ∫_{ℂ} ∫_{ℝ} K((z, t), (w, s))f(w, s) dλ(w)ds.

Then, for any f ∈ C_{0}^{∞}(ℝ^{3}), T f satisfies LT f = f − Sf and T f⊥Ker(L). Here, S is the Szegö projection.

The kernel K is homogeneous of degree −2k − 1 with respect to the dilation δ(z, t) = (δz, δ^{2k}t) on ∂Ω_{p}. This gives rise to define a nonisotropic pseudometric (see [D])

d((z, t), (w, s)) = |A^{1/k} − z\overline{w}|^{1/2}

on ∂Ω_{p}, in order obtain suitable estimates of the kernel K. Using a method for homogeneous kernel functions on the Heisenberg group due to Folland and Stein [FS] one obtains

Theorem 4 ([K]). Let ω(z, t) = (1 + d((0, 0), (z, t)))^{−2} . If f ∈ L^{2}(ℝ^{3}) satisfies Sf = 0, then there exists a unique u ∈ L_{ω}^{2}(ℝ^{3}) such that Lu = f and ∥u∥_{2,ω} ≤ ∥f∥_{2}. This means that the relative fundamental solution operator T is bounded as an operator form (H^{2}(∂Ω_{p}))^{⊥} to L_{ω}^{2}(ℝ^{3}).

In the second case p(z) = x^{2m}, where z = x+iy and m ∈ ℕ, it is much more difficult to find an apropriate expression for the kernel function of the relative fundamental solution. Domains of this type are of importance in finding counterexamples to analytic hypoellipticity (see [CG]).

Theorem 5 ([Z]). Let z = x + iy and w = u + iv and K((x, y, t), (u, v, s))

= ∫_{0}^{∞} e^{2πτ(u^{m}−x^{m}−i(t−s))} ∫_{−∞}^{∞} e^{2πη(u−x+i(y−v))} ∫_{−∞}^{u} \frac{e^{−4π(ηζ+τζ^{m})}}{∫_{−∞}^{∞} e^{−4π(ηs+τs^{m})} ds} dζ dη dτ

+ ∫_{0}^{∞} e^{−2πτ(u^{m}−x^{m}−i(t−s))} ∫_{−∞}^{∞} e^{2πη(u−x+i(y−v))} ∫_{x}^{∞} \frac{e^{4π(ηζ−τζ^{m})}}{∫_{−∞}^{∞} e^{4π(ηs−τs^{m})} ds} dζ dη dτ.

Define an operator T by

T f(x, y, t) = ∫_{ℝ} ∫_{ℝ} ∫_{ℝ} K((x, y, t), (u, v, s))f(u, v, s) dudvds.

Then, for any f ∈ C_{0}^{∞}(ℝ^{3}), T f satisfies LT f = f − Sf and T f⊥Ker(L). Here, S is the Szegö projection.

As before, the kernel K is homogeneous of degree −2m − 1 with respect to the dilation δ(x, y, t) = (δx, δy, δ^{2m}t) on ∂Ω_{p}. In this case the corresponding nonisotropic pseudometric d((z, t), (w, s)) is defined to be the infimum of all δ > 0 satisfying

|(\frac{x^{m} + u^{m}}{2} − i\frac{t − s}{2}) − (\frac{z + \overline{w}}{2} )^{m}| < δ^{2} |\frac{z + \overline{w}}{2} |^{m−2} + δ^{m}

and |z − w| < δ. Now it is quite complicated to estimate the kernel K using the above pseudometric, but one gets an analogous result

Theorem 6 ([Z]). Let ψ_{q}(z, t) = (1 + d((0, 0), (z, t)))^{−q} . If f ∈ L^{q}(ℝ^{3}) satisfies Sf = 0, then there exists a unique u ∈ L_{ψ_{q}}^{q}(ℝ^{3}) such that Lu = f and ∥u∥_{q,ψ_{q}} ≤ ∥f∥_{q}, where 1 < q < m + 2. For q = 2 this means that the relative fundamental solution operator T is bounded as an operator form (H^{2}(∂Ω_{p}))^{⊥} to L_{ψ_{2}}^{2}(ℝ^{3}).

For further details see [Z].

3. Bergman and Szegö kernels.

Now 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} = {(z, w) : z ∈ ℂ^{n} , w ∈ ℂ , ℑw > p(z)} :

Theorem 7.

(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.

For the proof see [Has3].

4. Applications.

First we compute or at least estimate the Bergman kernel B of certain unbounded domains Ω_{p} in the point ((0, i), (0, i)) , 0 ∈ ℂ^{n}, of the diagonal. Applying a theorem of Boas, Straube and Yu [BSY] this determines the boundary behavior of the Bergman kernel K_{Ω} on the diagonal of a bounded pseudoconvex domain Ω in ℂ^{n+1} that is h-extendible at the boundary point Q with multiple type (m_{0}, m_{1},…, m_{n}) and that has Ω_{p} as local model at Q :

lim_{\begin{matrix}z→Q \\ z∈Γ \end{matrix}} K_{Ω}(z) d(z)^{∑_{j=0}^{n} 2/m_{j}} = B((0, i), (0, i)),

here Γ is a nontangential cone in Ω with vertex at Q and d(z) is the distance from z to the boundary of Ω.

Theorem 8.

B((0, i), (0, i)) = 4π ∫_{0}^{∞} τK_{τ} (0, 0)e^{−4πτ} dτ,

where K_{τ} denotes the Bergman kernel of the Hilbert space H_{τ} of entire functions ϕ : ℂ^{n} ⟶ ℂ, such that

∫_{ℂ^{n}} |ϕ(z)|^{2}e^{−4πτp(z)} dλ(z) , τ > 0.

Finally we compute B((0, i), (0, i)) for two classes of examples. Here we concentrate on the so-called decoupled domains (see [McN2]).

Theorem 9. If p(z_{1},…, z_{n}) = ∑_{j=1}^{n} |z_{j}|^{α_{j}} , where α_{j} > 1 , j = 1,…, n, then

B((0, i), (0, i)) = \frac{Γ (2 + \frac{2}{α_{1}} + ⋯ + \frac{2}{α_{n}} ) ∏_{j=1}^{n} α_{j}}{2^{n+4}π^{n+2} ∏_{j=1}^{n} Γ (\frac{2}{α_{j}} )} .

Theorem 10. If p(z_{1},…, z_{n}) = ∑_{j=1}^{n} \frac{|x_{j}|^{α_{j}}}{α_{j}} , where z_{j} = x_{j} +iy_{j} , α_{j} > 1 , j = 1,…, n, then the Bergman kernel in the point ((0, i), (0, i)) can be estimated in the form

(4π)^{−n−1} (\frac{8}{27})^{n} Γ (2 + \frac{2}{α_{1}} + ⋯ + \frac{2}{α_{n}} ) ∏_{j=1}^{n} [α_{j}^{′−1+2/α′_{j}}Γ (\frac{2}{α′_{j}} )]

≤ B((0, i), (0, i))

≤ (4π)^{−n−1}Γ (2 + \frac{2}{α_{1}} + ⋯ + \frac{2}{α_{n}} ) ∏_{j=1}^{n} [γ_{j}α_{j}^{′−1+1/α′_{j}}Γ (\frac{1}{α′_{j}} )] ,

with 1/α_{j} + 1/α′_{j} = 1 and γ_{j} = min_{0<μ<1}{2e^{χ_{j}(μ_{j})}/μ_{j}}, where

χ_{j}(μ_{j}) = β_{j} [(α_{j}β_{j})^{1/(α_{j}−1)} − 1]^{−(α_{j}−1)} , β_{j} = 1 − \frac{μ_{j}^{α′_{j}}}{α′_{j}} .

For further details see [Has3].

Next 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 indicated that these kernel functions have no zeroes in Ω_{p}.

Theorem 11. 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.

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 [BFS]). Using the above formula for the Bergman kernel function one obtains the following result

Theorem 12. 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}.

See [Has4] for the details.

5. The Bergman kernel and a generalized Fourier–Borel transform.

We describe here, how the Bergman kernel of weighted spaces of entire functions we considered above can be used to establish dual representations of Fréchet spaces of entire functions. In the classical setting the exponential functions provide the isomorphism via Fourier-Borel transform. In our case we use the Bergman kernel instead of the exponential functions. Let P = (p_{m})_{m≥1} be a decreasing sequence of continuous functions p_{m} : ℂ^{n} ⟶ ℝ. We consider spaces F_{P} of entire functions the topology of which is given by a sequence of Hilbert norms

∥f∥_{m} = [∫_{ℂ^{n}} |f(z)|^{2} exp(−2p_{m}(z)) dλ(z)]^{1/2} ,

where λ denotes the Lebesgue measure on ℂ^{n}. Suppose that the function exp(2p_{1}) is locally integrable. In this way F_{P} becomes a Fréchet space. The topology of F_{P} is in general stronger than the topology of uniform convergence on compact subsets of ℂ^{n}. F_{P} is in fact the intersection of the Hilbert spaces H_{p_{m}} of entire functions f such that ∥f∥_{m} < ∞. Each of the Hilbert spaces H_{p_{m}} possesses a reproducing kernel. We will describe the dual space F′_{P} again as a space of entire functions. For this purpose it is essential that the Hilbert space H_{p_{0}} is large enough, where p_{0}(z) = lim_{m→∞} p_{m}(z).

The Bergman kernel of the space H_{p_{0}} will be used to obtain the weight functions p_{m}^{∗} of the dual Fréchet space of entire functions which will turn out to be isomorphic to the dual space F′_{P} . In the classical setting the exponential functions z ↦ exp < z, w > are used to establish the isomorphism by the Fourier–Borel transform, but for this purpose the exponential functions z ↦ exp < z, w > have to belong to the original space F_{P} . In our case we use the Bergman kernel K_{p_{0}}(z, w) of the space H_{p_{0}} instead of the exponential functions in order to establish the isomorphism. The weight functions p_{m}^{∗} defined by

exp(2p_{m}^{∗}(w)) = ∫_{ℂ^{n}} |K_{p_{0}}(z, w)|^{2} exp(−2p_{m}(z)) dλ(z),

correspond to the Young conjugates of the weights p_{m} (see [Has5]). We use several properties of the Bergman kernel and have to impose a certain growth condition on the Bergman kernel K_{p_{0}}(z, w) which makes the calculations work. This growth condtion is always satisfied in the case of radial weight functions p_{m}(z) = r_{m}|z|^{α}, where α > 0 and (r_{m})_{m} is a strictly decreasing sequence of positive real numbers, such that lim_{m→∞} r_{m} = r_{0} > 0. A different approach to similar results can be found in [LG].

Weighted spaces of entire functions appear in a natural way in the context of analytically uniform spaces ([BD], [BT], [E], [Ma1], [Ma2]), where the problem of finding a distributional solution of a partial differential equation, via Fourier–Borel transform, can be viewed as a problem of mappings between various weighted spaces of entire functions. Reproducing kernels in Hilbert spaces of entire functions, for example the Bargmann-Fock space, are also investigated in [Sa].

The dual representation of F_{P} is described in the following way

Theorem 13. Assume that H_{p_{0}} is dense in F_{P} and that for each m ∈ ℕ there exists a number l ∈ ℕ such that

(*)  ∫_{ℂ^{n}} ∫_{ℂ^{n}} |K_{p_{0}}(z, w)|^{2} exp(−2p_{m}(z) − 4p_{0}(w) + 2p_{l}(w)) dλ(z) dλ(w) < ∞.

Then the dual space F′_{P} , endowed with the strong topology, is topologically isomorphic to the dual Fréchet space G_{P^{∗}} of entire functions, where

G_{P^{∗}} = ⋃_{m=1}^{∞} G_{p_{m}},

where

G_{p_{m}} = {F : ℂ^{n} ⟶ ℂ entire : ∥F ∥_{m}^{∗} = sup{|F (w)| exp(−p_{m}^{∗}(w)) : w ∈ ℂ^{n}} < ∞}

the isomorphism being given by L ↦ L̂ , L ∈ F′_{P} . In addition the following inequality holds : for each m ∈ ℕ there exists a number l ∈ ℕ and a constant C = C(l, m) > 0 such that

C∥L∥′_{l} ≤ ∥L̂∥_{m}^{∗} ≤ ∥L∥′_{m},

for L ∈ F′_{P} , where

∥L∥′_{m} = sup{|L(f)| : f ∈ F_{P} , ∥f∥_{m} ≤ 1}.

Remarks.

(a) Condition (*) is in a certain sense a sharpened form of the nuclearity of the natural imbedding

I_{0,m} : H_{p_{0}} ⟶ H_{p_{m}}.

The operator I_{0,m} is nuclear if and only if

∫_{ℂ^{n}} K_{p_{0}}(z, z) exp(−2p_{m}(z)) dλ(z) < ∞

(compare [Has6]). Let l → ∞ in (*), then

∫_{ℂ^{n}} ∫_{ℂ^{n}} |K_{p_{0}}(z, w)|^{2} exp(−2p_{m}(z) − 2p_{0}(w)) dλ(z) dλ(w)

= ∫_{ℂ^{n}} K_{p_{0}}(z, z) exp(−2p_{m}(z)) dλ(z).

(b) Integration with respect to z in (*) does not cause any troubles, because

K_{p_{0}}(., w) ∈ H_{p_{m}} for any w ∈ ℂ^{n},

but integration with respect to w does, because

exp(−4p_{0}(w) + 2p_{l}(w)) ≥ exp(−2p_{0}(w)).

In this connection it is reasonable to restrict on weight functions p_{m} such that p_{m}(z) ≤ 2p_{0}(z), for each z ∈ ℂ^{n} and m ∈ ℕ, because then −4p_{0}(z) + 2p_{m}(z) ≤ 0. We mention that the estimate

|K_{p_{0}}(z, w)|^{2} ≤ K_{p_{0}}(z, z)K_{p_{0}}(w, w)

is too coarse to verify condition (*).

In the following situation it can be shown that assumptions of Theorem 13 are satisfied: (in sake of simplicity we now restrict our considerations to the case n = 1, for n > 1 analogous results hold.)

Theorem 14. Let p_{m}(z) = r_{m}|z|^{α}, where α > 0 and r_{m} ↓ r_{0} > 0. Then the dual space F′_{P} can be identified with the dual Fréchet space G_{P^{∗}} of entire functions F such that

∥F ∥_{m}^{∗} = sup \{|F (w)| exp (−\frac{r_{0}^{2}}{r_{m}} |w|^{α}) : w ∈ ℂ\} < ∞,

for some m ∈ ℕ.

See [Has6] for the details.

Remarks. We point out that the case 0 < α ≤ 1 is of special interest, because in this case the Fourier–Borel transform cannot be applied.

For a fixed weight function p and the Bergman kernel K_{τ} (z, w) of the Hilbert space of entire functions f : ℂ^{n} ⟶ ℂ such that

∫_{ℂ^{n}} |f(z)|^{2} exp(−2τp(z)) dλ(z) < ∞,

for τ > 0, condition (*) corresponds to the following property: for each σ > τ there exists a number μ > 0 with 0 < μ < τ such that

∫_{ℂ^{n}} ∫_{ℂ^{n}} |K_{τ} (z, w)|^{2} exp(−2σp(z) − 2μp(w))) dλ(z) dλ(w) < ∞,

which is an interesting property in itself.

References

[BFS] H.P. Boas, S. Fu and E.J. Straube, The Bergman kernel function: Explicit formulas and zeroes, Proc. Amer. Math. Soc. (to appear).

[B] A. Boggess, CR manifolds and the tangential Cauchy-Riemann complex, CRC Press, Boca Raton, Ann Arbor, London, Tokyo, 1992.

[BD] C.A. Berenstein and M.A. Dostal, Analytically uniform spaces and their applications to convolution equations, Lecture Notes in Mathematics, Vol. 256, Springer Verlag, Berlin, 1972.

[BSY] H.P. Boas, E.J. Straube and J. Yu, Boundary limits of the Bergman kernel and metric, Michigan Math.J. 42 (1995), 449–462.

[BT] C.A. Berenstein and B.A. Taylor, A new look at interpolation theory for entire functions of one variable, Adv. in Math. 33 (1979), 109–143.

[CG] M. Christ and D. Geller, Counterexamples to analytic hypoellipticity for domains of finite type, Ann. of Math. 135 (1992), 551–566.

[D’A] J.P. D’Angelo, An explicit computation of the Bergman kernel function, J. of Geom. Analysis 4 (1994), 23–34.

[D] K. P. Diaz, The Szegö kernel as a singular integral kernel on a family of weakly pseudoconvex domains, Trans. Amer. Math. Soc. 304 (1987), 147–170.

[DO] K. Diederich and T. Ohsawa, On the parameter dependence of solutions to the \overline{∂}-equation, Math. Ann. 289 (1991), 581–588.

[E] L. Ehrenpreis, Fourier analysis in several complex variables, Wiley–Interscience, New York, 1970.

[FH1] G. Francsics and N. Hanges, Explicit formulas for the Szegö kernel on certain weakly pseudoconvex domains, Proc. Amer. Math. Soc. 123 (1995), 3161–3168.

[FH2] G. Francsics and N. Hanges, The Bergman kernel of complex ovals and multivariable hypergeometric functions, J. of Functional Analysis 142 (1996), 494–510.

[FH3] G. Francsics and N. Hanges, Asymptotic behavior of the Bergman kernel and hypergeometric functions, Contemporary Math. (to appear).

[GS] P. C. Greiner and E. M. Stein, On the solvability of some differential operators of type □_{b}, Proc. Internat. Conf., (Cortona, Italy, 1976–1977), Scuola Norm. Sup. Pisa (1978), 106–165.

[Han] N. Hanges, Explicit formulas for the Szegö kernel for some domains in ℂ^{2}, J. Functional Analysis 88 (1990), 153–165.

[Has1] F. Haslinger, Szegö kernels of certain unbounded domains in ℂ^{2}, Révue Roum. Math. Pures et Appl. 39 (1994), 914–926.

[Has2] F. Haslinger, Singularities of the Szegö kernel for certain weakly pseudoconvex domains in ℂ^{2}, J. Functional Analysis 129 (1995), 406–427.

[Has3] F. Haslinger, Bergman and Hardy Spaces on model domains, Ill. J. Math. 42 (1998), 458–469.

[Has4] F. Haslinger, The Bergman kernel functions for certain unbounded domains, Ann. Polon. Math. (to appear).

[Has5] F. Haslinger, Weighted spaces of entire functions, Indiana Univ. Math. J. 35 (1986), 193–207.

[Has6] F. Haslinger, The Bergman kernel and a generalized Fourier–Borel transform, Proceedings of the ISAAC Conference 1997, Kluwer (to appear).

[K] H. Kang, \overline{∂}_{b}–equations on certain unbounded weakly pseudoconvex domains, Trans. Amer. Math. Soc. 315 (1989), 389–413.

[Kam] J. Kamimoto, Asymptotic expansion or the Bergman kernel for weakly pseudoconvex tube domains in ℂ^{2}, Preprint series, Graduate School of Mathematical Sciences, University of Tokyo 96-4 (1996).

[LG] P. Lelong and L. Gruman, Entire functions of several complex variables, Springer–Verlag, Grundlehren der mathematischen Wissenschaften Vol. 282, Berlin–New York, 1986.

[Ma1] A. Martineau, Sur les fonctionelles analytiques et la transformation de Fourier-Borel, J.d’Analyse Math. 9 (1963), 1–144.

[Ma2] A. Martineau, Equations différentielles d’ordre infini, Bull. Soc. math. France 95 (1967), 109–154.

[McN1] J. McNeal, Boundary behavior of the Bergman kernel function in ℂ^{2}, Duke Math. J. 58 (1989), 499–512.

[McN2] J. McNeal, Local geometry of decoupled pseudoconvex domains, Complex Analysis, Aspects of Math., K. Diederich, ed., Vieweg E17 (1991), 223–230.

[M] W. Müller, Randverhalten holomorpher Funktionen auf gewissen schwach pseudokonvexen Gebieten, Dissertation, Universität Wien, 1994.

[N] A. Nagel, Vector fields and nonisotropic metrics, Beijing Lectures in Harmonic Analysis, (E. M. Stein, ed.), Princeton University Press, Princeton, NJ, 1986, pp. 241–306.

[NRSW1] A. Nagel, J. P. Rosay, E. M. Stein and S. Wainger, Estimates for the Bergman and Szegö kernels in certain weakly pseudoconvex domains, Bull. Amer. Math. Soc. 18 (1988), 55–59.

[NRSW2] A. Nagel, J. P. Rosay, E. M. Stein and S. Wainger, Estimates for the Bergman and Szegö kernels in ℂ^{2}, Ann. of Math. 129 (1989), 113–149.

[OPY] K. Oeljeklaus, P. Pflug and E. H. Youssfi, The Bergman kernel of the minimal ball and applications, Ann. Inst. Fourier 47 (1997), 915–928.

[Ra] M. Range, Holomorphic functions and integral representations in several complex variables, Springer Verlag, 1986.

[Sa] S. Saitoh, Integral transforms, reproducing kernels and their applications, Addison, Pitman Research Notes in Mathematics, Vol. 369, 1997.

[S] E.M. Stein, Harmonic Analysis. Real-variable methods, orthogonality and oscillatory integrals, Princeton University Press, Princeton, NJ, 1993.

[SW] E.M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, Princeton, NJ, 1971.

[Z] L. Zinner, A realtive fundamental solution for the Lewy operator, regularity of the Szegö projection and the solution operator, Dissertation, Universität Wien, 1998.

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