Bergman and Hardy spaces on model domains.

Friedrich Haslinger*

1. Introduction

Let p : ℂ ⟶ ℝ_{+} denote a C^{1}–function and define Ω_{p} ⊆ ℂ^{2} by

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

Weakly pseudoconvex domains of this kind were investigated by McNeal [McN1] 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, 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 kernel function on various weakly pseudoconvex domains ( [D’A], [BFS], [FH2] and [FH3] ).

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 32A35; Secondary 30D20, 32F40.

Key words and phrases. Hardy spaces, Hilbert spaces of entire functions, model domains, CR functions, Bergman kernel, Szegö kernel.

*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 main result of this paper is to show 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. In fact the space 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 distribution sense (Theorem 1). We also point out how this result can be extended to corresponding domains in ℂ^{n+1}.

In the next result we express the Bergman and the Szegö kernel of the domains Ω_{p} by means of the Bergman kernel of certain weighted Hilbertspaces of entire funtions (Theorem 2 and Theorem 3).

Finally we apply these results to determine the boundary limits of the Bergman kernel on the diagonal of a bounded pseudoconvex domain Ω in ℂ^{n+1}, that is h-extendible at the boundary point P, using a reduction to the model case due to Boas, Straube and Yu [BSY] (Theorem 4 and Theorem 5).

2. Weighted spaces of entire functions with parameters

In sake of simplicity we concentrate on domains Ω_{p} in ℂ^{2}, see Remark (b) after Theorem 1 for domains in higher dimensions.

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 lemma is a version of an important representation result for Hardy spaces (see [SW] for a special case).

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

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

where F ∈ E_{p}. In addition F can be recovered from the boundary value of f by

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

Proof. 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

(3)  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) . Then from (1) we get

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

If f ∈ H^{2}(∂Ω_{p}), then its boundary function f_{0} satisfies \overline{L}(f_{0}) = 0 in the sense of distributions (the functions f_{y} are holomorphic in a neighborhood of ∂Ω_{p}, they satisfy the equation \overline{L}(f_{y}) = 0 (see [Ra]) and they converge to the boundary function f_{0} in L^{2}). Now let F be as in (2). Then, using Plancherel’s theorem, we get

F^{−1}M^{−1}F = f_{0}

and from (4)

0 = \overline{L}(f_{0}) = \overline{L}F^{−1}M^{−1}F = F^{−1}M^{−1} \frac{∂}{∂\overline{z}} F,

which implies that \frac{∂}{∂\overline{z}} F = 0. Again by Plancherel’s theorem we obtain

∫_{0}^{∞} |F (z, τ)|^{2}e^{−4πτp(z)} dτ = ∫_{ℝ} |f_{0}(z, t)|^{2} dt = ∫_{ℝ} |f(z, t + ip(z))|^{2} dt,

hence

∫_{ℂ} ∫_{0}^{∞} |F (z, τ)|^{2}e^{−4πτp(z)} dτdλ(z) = ∫_{ℂ} ∫_{ℝ} |f(z, t + ip(z))|^{2} dtdλ(z).

Since

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

we get F ∈ E_{p}.

For (z, w) ∈ Ω_{p} we set

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

we can write w = t + i(p(z) + ρ) , t ∈ ℝ , ρ > 0 and from F ∈ E_{p} we derive that h is holomorphic in Ω_{p} (see for instance [Ru], pg. 404). A further application of Plancherel’s theorem implies that h = f. □

2. Extension of CR-distributions

Theorem 1. 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.

Proof. We define a function F on ℂ × ℝ_{+} by formula (2)

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

and conclude again from (4) 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. □

Remarks:

a) If f belongs to H^{2}(∂Ω_{p}), then its boundary value f_{0} satisfies \overline{L}(f_{0}) = 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.

b) If Ω_{p} is a domain in ℂ^{n+1} of the form

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

then the tangential Cauchy-Riemann operator has the form

\overline{∂}_{b}(f) = ∑_{j=1}^{n} \overline{L}_{j}(f) d\overline{z}_{j},

where

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

The same reasoning as in Theorem 1 applies now to these domains Ω_{p} in ℂ^{n+1}.

c) The above extension of CR-distributions to the whole domain Ω_{p} seems to be new even in the case of the Siegel upper half space (p(z) = |z|^{2}), see for instance [S] pg.642, where only a local extension property is mentioned. Theorem 1 gives the global extension in one step.

3. Bergman and Szegö kernels on certain unbounded domains.

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} :

Theorem 2.

(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.

Proof. (a) and (b) follow directly from Theorem 1 in [Has1] since the weight function satisfies the assumptions there. Special cases can be found in [GS] and [FH1]. The proof of Theorem 1 in [Has1] uses again the operator identity

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

For (c) one has to recall the formula

B((z′, z), (w′, w)) = 2i\frac{∂S}{∂\overline{w}} ((z′, z), (w′, w))

from [NRSW2]. □

4. Boundary limits of the Bergman kernel.

In this part we compute or at least estimate the Bergman kernel B of certain unbounded domains Ω_{p} at 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 3.

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.

See Theorem 2(c).

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

a) p(z_{1},…, z_{n}) = ∑_{j=1}^{n} |z_{j}|^{α_{j}} , where α_{j} > 1 , j = 1,…, n. In this (radial) case the Bergman kernel of the weighted spaces of entire functions H_{τ} can be represented by the orthonormalized system of the monomials and a computation using the formula

(5)  ∫_{0}^{∞} τ^{m} exp(−cτ^{q}) dτ = \frac{Γ (\frac{m+1}{q})}{qc^{(m+1)/q}} , c, m, q > 0,

(see [GR]) proves the following

Theorem 4. 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}} )} .

b) 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. In this (nonradial) case the computation of the Bergman kernel is more complicated. We use a formula from [Has1] for the Bergman kernels K_{τ} of the weighted spaces of entire functions H_{τ} : for r = (r_{1},…, r_{n}) , r_{j} ∈ ℝ, we write p(r_{1},…, r_{n}) = ∑_{j=1}^{n} p_{j}(r_{j}) and p_{j}(r_{j}) = \frac{|r_{j}|^{α_{j}}}{α_{j}} , then

K_{τ} (z, w) = ∏_{j=1}^{n} ∫_{ℝ} \frac{exp(2πη_{j}(z_{j} + \overline{w}_{j}))}{∫_{ℝ} exp(4π(r_{j}η_{j} − τp_{j}(r_{j})) dr_{j}} dη_{j}.

In order to apply Theorem 3 we have to compute K_{τ} (0, 0). We consider a single factor in the above formula for K_{τ} (z, w) and omit the index j. Let

I(η, τ) = [∫_{ℝ} exp(4π(ηr − τp(r)) dr]^{−1} .

A straightforward computation shows that

∫_{ℝ} I(η, τ) dη = (4π)^{1/α−1/α′}τ^{2/α} ∫_{ℝ} \frac{dη}{∫_{ℝ} exp(rη − p(r)) dr} ,

where 1/α + 1/α′ = 1 We set p^{∗}(s) = |s|^{α′}/α′, which is the Young conjugate of p. In the following we estimate

∫_{ℝ} exp(rη − p(r)) dr

from below and from above in terms of the Young conjugate: for this purpose let η > 0 and λ > 1, then we get from Young’s inequality

rη − p(r) ≤ rη − λrη + p^{∗}(λη)

and hence

∫_{0}^{∞} exp(rη − p(r)) dr ≤ \frac{exp(p^{∗}(λη))}{(λ − 1)η},

further we have

∫_{0}^{∞} exp(−rη − p(r)) dr ≤ \frac{1}{η} ,

and finally

∫_{ℝ} exp(rη − p(r)) dr ≤ \frac{(λ − 1) + exp(p^{∗}(λη))}{(λ − 1)η},

for η > 0. By similar argument we handle the case η < 0 and get

(6)  ∫_{ℝ} exp(rη − p(r)) dr ≤ \frac{(λ − 1) + exp(p^{∗}(λη))}{(λ − 1)|η|},

for η ≠ 0 and λ > 1 Next we claim that

(7)  ∫_{ℝ} exp(rη − p(r)) dr ≥ exp(−χ(μ)p^{∗}(μη)),

where 0 < μ < 1 and

χ(μ) = β [(αβ)^{1/(α−1)} − 1]^{−(α−1)} , β = 1 − \frac{μ^{α′}}{α′} .

In order to show (7) we first indicate that

p^{∗}(η) = max_{x≥0}\{x|η| − p(x)\} = \frac{|η|^{α′}}{α′} ,

and that the maximum is attained in the point

κ(η) = |η|^{1/(α−1)}.

Hence for η ≥ 0 it follows that

∫_{ℝ} exp(rη − p(r)) dr ≥ ∫_{0}^{∞} exp(rη − p(r)) dr ≥ exp[η(κ(η) + 1) − p(κ(η) + 1)]. The last expression can be written in the form

η(κ(η) + 1) − p(κ(η) + 1) = η^{α/(α−1)} + η − \frac{1}{α}(η^{1/(α−1)} + 1)^{α}.

On the other hand we have for 0 < μ < 1 :

μηκ(μη) − p(κ(μη)) = \frac{1}{α′} (μη)^{α′} = p^{∗}(μη)

and we show that

η^{α′} + η − \frac{1}{α}(η^{1/(α−1)} + 1)^{α} ≥ p^{∗}(μη) − χ(μ),

which gives inequality (7). For this aim we have to prove that

\frac{1}{α}(η^{1/(α−1)} + 1)^{α} − (1 − \frac{1}{α′} μ^{α′}) η^{α′} − η ≤ χ(μ).

We set β = 1 − \frac{1}{α′} μ^{α′}, then β > \frac{1}{α} and we have

\frac{1}{α}(η^{1/(α−1)} + 1)^{α} − βη^{α′} − η ≤ \frac{1}{α}(η^{1/(α−1)} + 1)^{α} − βη^{α′}.

A straightforward computation shows that the expression \frac{1}{α}(η^{1/(α−1)} + 1)^{α} − βη^{α′} attains its maximum when η = [(αβ)^{1/(α−1)} − 1]^{−(α−1)} and therefore we obtain

\frac{1}{α}(η^{1/(α−1)} + 1)^{α} − βη^{α′} ≤ β[(αβ)^{1/(α−1)} − 1]^{−(α−1)},

which proves (7). For η < 0 we can proceed in an analogous way arriving with the same inequality (7). From (6) and (7) we get for 0 < μ < 1 and λ > 1 :

(8) ∫_{ℝ} \frac{(λ − 1)|η|}{(λ − 1) + e^{p^{∗}(λη)}} dη ≤ ∫_{ℝ} \frac{dη}{∫_{ℝ} exp(rη − p(r)) dr} ≤ e^{χ(μ)} ∫_{ℝ} e^{−p^{∗}(μη)} dη.

For the left term of inequality (8) we mention that

\frac{(λ − 1)|η|}{λ}e^{−p^{∗}(λη)} ≤ \frac{(λ − 1)|η|}{(λ − 1) + e^{p^{∗}(λη)}} ,

for each η ∈ ℝ. Using formula (5) the integration of the term on the left side yields

\frac{(λ − 1)}{λ} ∫_{ℝ} |η|e^{−p^{∗}(λη)} dη = \frac{2(λ − 1)}{λ^{3}}α^{′−1+2/α′}Γ (\frac{2}{α′} ) ,

for each λ > 1 The maximal value of 2(λ − 1)/λ^{3} is 8/27. The right term of inequality (8) is equal to

\frac{2e^{χ(μ)}}{μ} α^{′−1+1/α′}Γ (\frac{1}{α′} ) .

It is easily seen that \frac{2e^{χ(μ)}}{μ} tends to infinity as μ tends to 0 or to 1 Now let 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}} .

Then

(4π)^{1/α_{j}−1/α′_{j}}τ^{2/α_{j}} \frac{8}{27}α_{j}^{′−1+2/α′_{j}}Γ (\frac{2}{α′_{j}} ) ≤ ∫_{ℝ} I_{j}(η_{j}, τ) tη

≤ (4π)^{1/α_{j}−1/α′_{j}}τ^{2/α_{j}}γ_{j}α_{j}^{′−1+1/α′_{j}}Γ (\frac{1}{α′_{j}} ) ,

which implies

∏_{j=1}^{n} [(4π)^{1/α_{j}−1/α′_{j}}τ^{2/α_{j}} \frac{8}{27}α_{j}^{′−1+2/α′_{j}}Γ (\frac{2}{α′_{j}} )] ≤ K_{τ} (0, 0)

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

In the last step we integrate over τ and obtain from Theorem 3:

Theorem 5. 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}} .

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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/f̃hasling