The Bergman kernel and a generalized Fourier–Borel transform.

Friedrich Haslinger*

Abstract. In this paper we represent the dual space of a Fréchet space of entire functions again as a space of entire functions. For this purpose we use the Bergman kernel of a certain Hilbert space. 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 in order to establish the isomorphism.

1. Introduction

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. These kernels are used to describe the adjoint operators of the inclusions H_{p_{l}} ↪ H_{p_{m}} (m > l) as certain Toeplitz operators mapping H_{p_{m}} to H_{p_{l}}. Now it is easy to derive a characterization of nuclearity of the space F_{P} in terms of the Bergman kernel (see [W]). The main topic of this paper is it to describe the dual space F′_{P} 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

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

Key words and phrases. Hilbert spaces of entire functions, Bergman kernel, Fourier–Borel transform.

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

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 [H1]). 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], [M1], [M2]), 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 [S]. They also appear in the context of Hardy and Bergman spaces on model domains (see [FH], [GS], [H2], [H3], [H4]).

2. Bergman kernels and Toeplitz operators

The inner product of the Hilbert space H_{p_{m}} is given by

(f, g)_{m} = ∫_{ℂ^{n}} f(z)\overline{g(z)} exp(−2p_{m}(z)) dλ(z),

for f, g ∈ H_{p_{m}}. The Fréchet space F_{P} is called nuclear if for each m ∈ ℕ there is a number l ∈ ℕ such that the canonical embedding

I_{l,m} : H_{p_{l}} ↪ H_{p_{m}}

is a nuclear operator (see [P] ). First we compute the adjoint operator of the canonical embedding I_{l,m}, which turns out to be a certain Toeplitz operator.

Proposition 1. The adjoint I_{l,m}^{∗} of I_{l,m} is a Toeplitz operator of the form

I_{l,m}^{∗}(g) = P_{l}(g exp(−2p_{m} + 2p_{l}),

where g ∈ H_{p_{m}} and P_{l} denotes the orthogonal projection of the Hilbert space

L_{p_{l}}^{2} = {f : ℂ^{n} ⟶ ℂ measurable : ∫_{ℂ^{n}} |f(z)|^{2} exp(−2p_{l}(z) dλ(z) < ∞}

onto the subspace H_{p_{l}} of all entire functions with the corresponding growth condition.

Proof. Since I_{l,m} is the canonical embedding we have

(I_{l,m}(f), g)_{m} = (f, g)_{m},

for f ∈ H_{p_{l}} and g ∈ H_{p_{m}}. Hence

(1)  (f, g)_{m} = (f, I_{l,m}^{∗}(g))_{l},

Now we consider the canonical embedding Ĩ_{l,m} : L_{p_{l}}^{2} ⟶ L_{p_{m}}^{2}

of the corresponding weighted L^{2}−spaces. From (1) we obtain

(2) ∫_{ℂ^{n}} f(z)\overline{g(z)} exp(−2p_{m}(z)) dλ(z) = ∫_{ℂ^{n}} f(z)[Ĩ_{l,m}^{∗}(g)(z)]^{−} exp(−2p_{l}(z)) dλ(z),

for f ∈ L_{p_{l}}^{2} and g ∈ L_{p_{m}}^{2}. Since f ↦ f exp(−p_{l}) is an isometry between L_{p_{l}}^{2} and L^{2} and since it follows from (2) that

∫_{ℂ^{n}} f(z) exp(−p_{l}(z))\overline{g(z)} exp(−2p_{m}(z) + p_{l}(z)) dλ(z)

= ∫_{ℂ^{n}} f(z) exp(−p_{l}(z))[Ĩ_{l,m}^{∗}(g)(z)]^{−} exp(−p_{l}(z)) dλ(z), (3)

for each f ∈ L_{p_{l}}^{2}, we obtain

(4)  Ĩ_{l,m}^{∗}(g) = g exp(−2p_{m} + 2p_{l}).

Hence for f ∈ H_{p_{l}} and g ∈ H_{p_{m}} we have by (4) and the fact that P_{l}(f) = f for f ∈ H_{p_{l}} :

(I_{l,m}(f), g)_{m} = (f, g)_{m} = (f, g exp(−2p_{m} + 2p_{l}))_{l}
= (P_{l}(f), g exp(−2p_{m} + 2p_{l}))_{l}

= (f, P_{l}(g exp(−2p_{m} + 2p_{l})))_{l}, (5) which proves Proposition 1.

We already mentioned that the topology of the Hilbert spaces H_{p_{m}} is in general stronger than the topology of uniform convergence on the compact subsets of ℂ^{n}. Especially, the point evaluations f ↦ f(z), z ∈ ℂ^{n} fixed, are continuous linear functional on H_{p_{m}}. Hence there exists a reproducing kernel K_{p_{m}}(z, w) for H_{p_{m}} i.e.

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

for each f ∈ H_{p_{m}}. K_{p_{m}}(z, w) is called the Bergman kernel of the Hilbert space H_{p_{m}} ; K_{p_{m}}(z, w) is conjugate symmetric, holomorphic in z and conjugate holomorphic in w.

Proposition 2. The operator I_{l,m}^{∗} can be written in the form

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

where g ∈ H_{p_{m}} and K_{p_{l}}(z, w) denotes the Bergman kernel of the Hilbert space H_{p_{l}}.

Proof. The orthogonal projection

P_{l} : L_{p_{l}}^{2} ⟶ H_{p_{l}}

is of the form

P_{l}(f)(z) = ∫_{ℂ^{n}} f(w)K_{p_{l}}(z, w) exp(−2p_{l}(w)) dλ(w),

for f ∈ L_{p_{l}}^{2}. Now the result follows directly from Proposition 1.

Using the fact that for an orthonormal basis (φ_{k})_{k} of H_{p_{l}} and for an arbitrary compact subset C ⊂ ℂ^{n} one has

∑_{k=1}^{∞} φ_{k}(z)\overline{φ_{k}(w)} = K_{p_{l}}(z, w),

uniformly on C × C (see [K], [R]), one gets

Proposition 3 ([W]). The Fréchet space F_{P} is nuclear if and only if for each m ∈ ℕ there exists a number l ∈ ℕ such that

(6)  ∫_{ℂ^{n}} K_{p_{l}}(w, w) exp(−2p_{m}(w)) dλ(w) < ∞.

Proposition 4. If the weight functions p_{m} satisfy the following two conditions:

(i) for each m ∈ ℕ there exists a number l ∈ ℕ such that

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

(ii)for each m ∈ ℕ there exists a number l ∈ ℕ and a constant C = C(m, l) > 0 such that

sup{p_{l}(z + w) : |w| ≤ 1} − p_{m}(z) ≤ C,

for each z ∈ ℂ^{n}, then the space F_{P} is nuclear.

This result can be derived from Proposition 3 (see [H1]).

3. Bergman kernels and duality

By construction, we have

H_{p_{0}} ⊆ ⋂_{m=1}^{∞} H_{p_{l}} = F_{P}

and

∥f∥_{0} = [∫_{ℂ^{n}} |f(z)|^{2} exp(−2p_{0}(z)) dλ(z)]^{1/2} ≥ ∥f∥_{m},

for each f ∈ H_{p_{0}} , m ∈ ℕ. We now take a continuous linear functional L ∈ F′_{P} and use the Bergman kernel K_{p_{0}}(z, w) of H_{p_{0}} in order to define an entire function L̂ related to L.

Lemma 1. Let L ∈ F′_{P} . Then

L̂ : w ↦ [L(K_{p_{0}}(., w))]^{−},

for w ∈ ℂ^{n}, is an entire function.

Proof. Since L ∈ F′_{P} , there exists m ∈ ℕ and a constant C_{m} > 0 such that

|L(f)| ≤ C_{m}∥f∥_{m},

for each f ∈ F_{P} . Hence

|L(f)| ≤ C_{m}∥f∥_{0},

for each f ∈ H_{p_{0}}, which means that L ∈ H′_{p_{0}}. Now let (φ_{k})_{k} be an orthonormal basis of H_{p_{0}}, then

K_{p_{0}}(., w) = ∑_{k=1}^{∞}[φ_{k}(w)]^{−}φ_{k},

with convergence in H_{p_{0}}. Therefore

[L(K_{p_{0}}(., w))]^{−} = ∑_{k=1}^{∞}[L(φ_{k})]^{−}φ_{k}(w).

All we have to show now is that the last series converges uniformly on compact subsets of ℂ^{n}. Since the sequence (L(φ_{k}))_{k} belongs to l^{2} this follows at once from

|∑_{k=1}^{∞}[L(φ_{k})]^{−}φ_{k}(w)|^{2} ≤ ∑_{k=1}^{∞} |L(φ_{k})|^{2} ∑_{k=1}^{∞} |φ_{k}(w)|^{2} = Const. K_{p_{0}}(w, w)

and from the fact that w ↦ K_{p_{0}}(w, w) is continuous (see [W]).

We now consider the space of all entire functions L̂ and define appropriate weight functions such that the dual sapce F′_{P} can be realized as a weighted dual Fréchet space of entire functions. We set

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

for m ∈ ℕ. Note that

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

Theorem 1. Assume that H_{p_{0}} is dense in F_{P} and that for each m ∈ ℕ there exists a number l ∈ ℕ such that

(7) ∫_{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 (7) 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 Proposition 3). Let l → ∞ in (7), 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 (7) 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 (7).

(c) Let H_{2p_{0}−p_{l}} denote the Hilbert space of all entire functions f such that

∥f∥_{−l} = [∫_{ℂ^{n}} |f(z)|^{2} exp(−4p_{0}(z) + 2p_{l}(z)) dλ(z)]^{1/2} < ∞.

If F_{P} is nuclear, it is easily seen that the natural embedding

H_{2p_{0}−p_{l}} ↪ H_{p_{0}}

is nuclear too. Using the spectral decompositon of this embedding, one can show that there exists on orthogonal basis (ψ_{k})_{k} in H_{2p_{0}−p_{l}} such that condition (7) can be expressed in the form

(7’)  ∑_{k=1}^{∞} ∥ψ_{k}∥_{m}^{2}∥ψ_{k}∥_{−l}^{2}∥ψ_{k}∥_{0}^{−4} < ∞.

Proof of Theorem 1. By Lemma 1, L̂ is an entire function. First we show that the mapping L ↦ L̂ is injective. For this aim take a continuous linear functional L ∈ F′_{P} such that

L(K_{p_{0}}(., w)) = 0

for each w ∈ ℂ^{n}. Now let (φ_{k})_{k} be an orthonormal basis of H_{p_{0}}. Since L is also an element of H′_{p_{0}}, we have

L(K_{p_{0}}(., w)) = ∑_{k=1}^{∞} L(φ_{k})[φ_{k}(w)]^{−} = 0,

for each w ∈ ℂ^{n} and since L corresponds to an element φ ∈ H_{p_{0}}, for which

φ = ∑_{k=1}^{∞}(φ, φ_{k})_{0}φ_{k} = ∑_{k=1}^{∞}[L(φ_{k})]^{−}φ_{k}

holds in H_{p_{0}}, we conclude that L = 0 on H_{p_{0}} and, as H_{p_{0}} is dense in F_{P} , L = 0 on F_{P} . Hence, L ↦ L̂ is injective. In order to show that this mapping is surjective, we take an arbitrary entire function F ∈ G_{P^{∗}} and show that there exists a continuous linear functional L ∈ F′_{P} such that L̂ = F. For F ∈ G_{P^{∗}} there exists a number m ∈ ℕ such that

∥F ∥_{m}^{∗} < ∞.

Now set

ψ(w) = F (w) exp(−2p_{0}(w)).

We claim that there exists a number l ∈ ℕ such that

∫_{ℂ^{n}} |ψ(w)|^{2} exp(2p_{l}(w)) dλ(w) < ∞.

To prove this we use condition (7) :

∫_{ℂ^{n}} |ψ(w)|^{2} exp(2p_{l}(w)) dλ(w) = ∫_{ℂ^{n}} |F (w)|^{2} exp(2p_{l}(w) − 4p_{0}(w)) dλ(w)

= ∫_{ℂ^{n}} |F (w)|^{2} exp(−2p_{m}^{∗}(w)) exp(2p_{m}^{∗}(w) + 2p_{l}(w) − 4p_{0}(w)) dλ(w)

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

Notice that F is also an element of H_{p_{0}}, because

∫_{ℂ^{n}} |F (w)|^{2} exp(−2p_{0}(w)) dλ(w) = ∫_{ℂ^{n}} |ψ(w)|^{2} exp(2p_{0}(w)) dλ(w)

≤ ∫_{ℂ^{n}} |ψ(w)|^{2} exp(2p_{l}(w)) dλ(w) < ∞.

Now we define

L(f) = ∫_{ℂ^{n}} f(z)\overline{ψ(z)} dλ(z),

for f ∈ F_{P} . Then L ∈ F′_{P} , because

|L(f)| = |∫_{ℂ^{n}} f(z) exp(−p_{l}(z))\overline{ψ(z)} exp(p_{l}(z)) dλ(z)|

≤ ∥f∥_{l} [∫_{ℂ^{n}} |ψ(z)|^{2} exp(2p_{l}(z)) dλ(z)]^{1/2} ;

and

[L̂(w)]^{−} = L(K_{p_{0}}(., w)) = ∫_{ℂ^{n}} K_{p_{0}}(z, w)\overline{ψ(z)} dλ(z)
= ∫_{ℂ^{n}} K_{p_{0}}(z, w)\overline{F (z)} exp(−2p_{0}(z)) dλ(z)
= [∫_{ℂ^{n}} F (z)K_{p_{0}}(w, z) exp(−2p_{0}(z)) dλ(z)]^{−}
= \overline{F (w)},

hence L̂ = F. It remains to show that the mapping L ↦ L̂ establishes a topological isomorphism between F′_{P} with the strong topology and G_{P^{∗}} with the inductive limit topology of the Banach spaces G_{p_{m}^{∗}} , m ∈ ℕ. This follows from the inequalities

∥L̂∥_{m}^{∗} = sup{|L̂(w)| exp(−p_{m}^{∗}(w)) : w ∈ ℂ^{n}}
= sup{|L(K_{p_{0}}(., w))| exp(−p_{m}^{∗}(w)) : w ∈ ℂ^{n}}
≤ ∥L∥′_{m} sup{∥K_{p_{0}}(., w)∥_{m} exp(−p_{m}^{∗}(w)) : w ∈ ℂ^{n}}
= ∥L∥′_{m}

and

∥L∥′_{l} = sup{|L(f)| : ∥f∥_{l} ≤ 1}
= sup {|∫_{ℂ^{n}} f(w)[L̂(w)]^{−} exp(−2p_{0}(w))dλ(w)| : ∥f∥_{l} ≤ 1}

= sup {|∫_{ℂ^{n}} f(w) exp(−p_{l}(w))[L̂(w)]^{−} exp(−2p_{0}(w) + p_{l}(w)) dλ(w)| : ∥f∥_{l} ≤ 1}

≤ [∫_{ℂ^{n}} |L̂(w)|^{2} exp(−4p_{0}(w) + 2p_{l}(w)) dλ(w)]^{1/2}

= [∫_{ℂ^{n}} |L̂(w)|^{2} exp(−2p_{m}^{∗}(w)) exp(2p_{m}^{∗}(w) − 4p_{0}(w) + 2p_{l}(w)) dλ(w)]^{1/2}

≤ ∥L̂∥_{m}^{∗} [∫_{ℂ^{n}} ∫_{ℂ^{n}} |K_{p_{0}}(z, w)|^{2} exp(−2p_{m}(z) − 4p_{0}(w) + 2p_{l}(w))dλ(z)dλ(w)]^{1/2} .

This proves Theorem 1.

In sake of simplicity we now restrict our considerations to the case n = 1, for n > 1 analogous results hold.

Proposition 5. 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.

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

Proof. In order to show that all assumptions of Theorem 1 are satisfied we use the fact that the linear span of the polynomials is dense in each of the Hilbert spaces H_{p_{m}}, m ∈ ℕ and m = 0. For α ≥ 1 this follows easily from [T] and for 0 < α < 1 it follows from the fact that the Taylor series expansion of each function f_{θ}, where f_{θ}(z) = f(θz), 0 < θ < 1, f ∈ H_{p_{m}}, is convergent in H_{p_{m}} and f_{θ} tends to f in H_{p_{m}} as θ tends to 1. Especially, H_{p_{0}} is dense in F_{P} . Since the weight functions p_{m} are radial, the monomials {1, z, z^{2},… } constitue an orthogonal basis in each of the Hilbert spaces H_{p_{m}} , m ∈ ℕ and m = 0. Using the integral formula for the Γ-function

Γ(u) = ∫_{0}^{∞} x^{u−1} e^{−x} dx,

ℜ(u) > 0, we obtain

∥z∥_{m}^{2} = \frac{2π}{α} (2r_{m})^{−2(n+1)/α} Γ(2(n + 1)/α),

m ∈ ℕ and m = 0. Since

K_{p_{m}}(z, w) = ∑_{k=0}^{∞} ∥z^{k}∥_{m}^{−2} z^{k}\overline{w}^{k},

we get

K_{p_{m}}(z, w) = \frac{α}{2π} ∑_{k=0}^{∞}(2r_{m})^{2(k+1)/α} [Γ(2(k + 1)/α)]^{−1} z^{k}\overline{w}^{k},

m ∈ ℕ and m = 0. Now it is easy to compute the weights p_{m}^{∗} :

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

= \frac{α^{2}}{(2π)^{2}} ∑_{k=0}^{∞} \{(2r_{0})^{4(k+1)/α} [Γ(2(k + 1)/α)]^{−2} 2π ∫_{0}^{∞} r^{2k+1} exp(−2r_{m}r^{α}) dr |w|^{2k}\}

= \frac{α}{2π} (\frac{2r_{0}^{2}}{r_{m}} )^{2/α} ∑_{k=0}^{∞} [Γ(2(k + 1)/α)]^{−1} [(\frac{2r_{0}^{2}}{r_{m}} )^{1/α} |w|^{α}]^{2k} .

It is not difficult to determine the asymptotic behavior of exp(2p_{m}^{∗}) (see [Bo]) :

exp(2p_{m}^{∗}(w)) ≍ exp (\frac{2r_{0}^{2}}{r_{m}} |w|^{α}) ,

as |w| → ∞. From this we can see that condition (7) of Theorem 1 is satisfied:

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

= ∫_{ℂ} exp(2p_{m}^{∗}(w) − 4p_{0}(w) + 2p_{l}(w)) dλ(w)

≤ Const. ∫_{ℂ} exp [(\frac{2r_{0}^{2}}{r_{m}} − 4r_{0} + 2r_{l}) |w|^{α}] dλ(w).

Choose l so large that

r_{l} < (2r_{m} − r_{0})r_{0}/r_{m},

this is possible, since r_{l} ↓ r_{0} and

r_{0} < (2r_{m} − r_{0})r_{0}/r_{m}.

Now, with this choice of l, we have

\frac{2r_{0}^{2}}{r_{m}} − 4r_{0} + 2r_{l} < 0,

and hence condition (7) is fulfilled.

Remark. For a fixed weight function p let K_{τ} (z, w) denote the Bergman kernel 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. Then condition (7) corresponds to the following property: for each σ > τ there exists a number μ > 0 with 0 < μ < τ such that

(7”)  ∫_{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.

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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/