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Revista de la Unión Matemática Argentina
versión impresa ISSN 0041-6932versión On-line ISSN 1669-9637
Rev. Unión Mat. Argent. v.46 n.1 Bahía Blanca ene./jun. 2005
The Bergman Kernel on Tube Domains
Ching-I Hsin,
Minghsin University of Science and Technology, Taiwan.
Abstract
Let be a bounded strictly convex domain in
, and
the tube domain over
. In this paper, we show that the Bergman kernel of
can be expressed easily by an integral formula.
2000 Mathematics Subject Classification : 32A07, 32A25.
Key words and phrases. Bergman kernel; Tube domain.
1. Introduction
Let be a domain. The Bergman kernel (see for instance [4])
is one of the important holomorphic invariants associated to
, but it is often difficult to compute
. We shall show that if
is a tube domain over a bounded strictly convex domain, then an easy computation leads quickly to its Bergman kernel.
![Ω ⊂ Rn](/img/revistas/ruma/v46n1/1a0315x.png)
![T = T Ω](/img/revistas/ruma/v46n1/1a0316x.png)
![]() | (1.1) |
known as the tube domain over . Let
be the Bergman kernel of
. The main result of this paper is as follows.
![TΩ](/img/revistas/ruma/v46n1/1a0321x.png)
![]() | (1.2) |
Given a function on
, we shall say that
reproduces
if
![]() | (1.3) |
![dV](/img/revistas/ruma/v46n1/1a0328x.png)
![T Ω](/img/revistas/ruma/v46n1/1a0329x.png)
![A2(TΩ)](/img/revistas/ruma/v46n1/1a0330x.png)
![h](/img/revistas/ruma/v46n1/1a0331x.png)
![TΩ](/img/revistas/ruma/v46n1/1a0332x.png)
![]() | (1.4) |
The Bergman kernel is uniquely characterized by the following three properties:
(i) for all
;
(ii) reproduces every element in
in the sense of (1.3);
(iii) for all
.
Our formulation of in (1.2) clearly satisfies condition (i). Therefore, to prove the theorem, we need to check conditions (ii) and (iii). This will be done by Propositions 2.4 and 2.5 in the next section.
2. The Bergman kernel on tube domains
Let be a bounded strictly convex domain in
, and let
be the tube domain as defined in (1.1). Since
is strictly convex,
is strictly pseudoconvex.
Let and
be defined as in (1.2). The Bergman space
consists of holomorphic functions
which satisfy (1.4). In this section, we show that
satisfies conditions (ii) and (iii) stated in the Introduction. Let
denote the polynomial functions, and we define
![-z2 -z2 P e = {p(z)e ; p(z) polynomial }.](/img/revistas/ruma/v46n1/1a0356x.png)
Lemma 2.1 reproduces every element of
.
![]() | (2.1) |
Here denote the usual dot product. In deriving this identity, we observe that both sides of (2.1) are holomorphic in
. Therefore, it suffices to check that it holds on the totally real subspace
. This can be obtained from standard integration tables ([3], 3.323
2).
Write on
, where
are variables on
respectively. Then
![∫ K(z, w)e- w2dV w∈TΩ ∫ ∫ 2 = (21π)n- (x,y)∈TΩ t∈Rn eit(z-(x-iy))γ-t 1e- (x+iy)dt dx dy --1--∫ ∫ ∫ 2 2 -1 = (2π)n Rn Ω( Rn exp(- itx - x - 2ixy)dx) exp(itz - ty + y )γt dy dt = --√1--∫ ∫ exp(- (t+2y)2)exp(itz - ty + y2)γ-1dy dt by (2.1) (2 π)n∫Rn ∫Ω 4 t = --√1-n n exp(- t2- 2ty + itz)γ-t 1dy dt (2 π) ∫R Ω 2 4 = (2√1π)n Rn exp(- t4 + itz)dt -z2 = e . by (2.1)](/img/revistas/ruma/v46n1/1a0368x.png)
![]() | (2.2) |
From (2.2), substitute with
, where
. This gives
![1 ∫ ∫ eit(z+c-w¯) 2 2 ----n- ---------e- w dtdV = e-(z+c). (2 π) w∈TΩ t∈Rn γt](/img/revistas/ruma/v46n1/1a0373x.png)
Changing the variable to
in LHS gives
![1 ∫ ∫ eit(z- ¯w) ------ -------e-(w+c)2dt dV = e-(z+c)2. (2π)n w∈TΩ t∈Rn γt](/img/revistas/ruma/v46n1/1a0376x.png)
Apply to both sides, we see that
reproduces the function
. We carry out the procedure
for
, and Lemma 2.1 follows.
We plan to show that reproduces every element of
. In view of Lemma 2.1, this will follow if we can show that
is a dense subset of the Hilbert space
. This will be established by the next two lemmas. Our strategy is to convert the problem on
to another Hilbert space
by Lemma 2.2, and obtain a result on
by Lemma 2.3. We then transfer this result back to
, by Proposition 2.4.
Let denote the Hilbert space of
functions on
, with weight
given in (1.2). Namely,
![∫ 2 n 2 L (R ,γt) = {f ; Rn ∣f (t)∣ γtdt < ∞}.](/img/revistas/ruma/v46n1/1a0395x.png)
Lemma 2.2 (T. G. Genchev) The transformation
![]() | (2.3) |
is an isometry from to
, preserving the Hilbert space norms.
![h(z) ∈ A2(T Ω)](/img/revistas/ruma/v46n1/1a0399x.png)
![g(t) ∈ L2(Rn, γt)](/img/revistas/ruma/v46n1/1a03100x.png)
![]() | (2.4) |
see [1],[2]. This means that the transformation in (2.3) is surjective. In order to see that (2.3) is well-defined, injective and preserves the norms, we shall prove that in (2.4), .
Write , and (2.4) gives
![∫ -ixt -yt h(- x + iy) = n e (e g(t))dt. R](/img/revistas/ruma/v46n1/1a03104x.png)
Thus for every fixed ,
is the Fourier transform of
. By Plancherel's theorem,
![∫ ∫ ∣h(- x + iy)∣2dx = ∣e-ytg(t)∣2dt. Rn Rn](/img/revistas/ruma/v46n1/1a03108x.png)
Apply to both sides, we get
![∫ ∫ ∥h∥2A2(TΩ) = Ω Rn ∣h( - x + iy)∣2dx dy ∫ ∫ - yt 2 = Ω Rn ∣e g(t)∣ dtdy = ∫ ∣g(t)∣2γ dt Rn t = ∥g ∥22 n . L (R ,γt)](/img/revistas/ruma/v46n1/1a03110x.png)
This proves Lemma 2.2. □
Next we define
![- t2- - t2 Pe 4 = {p(t)e 4 ; p(t) polynomial }.](/img/revistas/ruma/v46n1/1a03112x.png)
Lemma 2.3 is dense in the Hilbert space
.
![]() | (2.5) |
is a complete basis of , and the lemma follows.
![f(t) ∈ L2(Rn, γt)](/img/revistas/ruma/v46n1/1a03117x.png)
![]() | (2.6) |
for all . It is easy to see that
is bounded, so
![∫ - t2 2 ∫ 2 - t2-2 Rn ∣e 4 f(t)γt∣ γtdt = Rn ∣f(t)∣γt(e 2 γt) dt ≤ c ∫ ∣f (t)∣2γ dt Rn t < ∞, as f(t) ∈ L2(Rn, γt).](/img/revistas/ruma/v46n1/1a03121x.png)
![- t2 2 n e 4 f(t)γt ∈ L (R ,γt)](/img/revistas/ruma/v46n1/1a03122x.png)
![]() | (2.7) |
By Lemma 2.2, . We may assume that
, and consider the power series expansion
![∑ k h(z) = akz](/img/revistas/ruma/v46n1/1a03126x.png)
near . Then
![ak = k1!h(k)(0) -1 ∣k∣∫ k - t2 = k!i Rn t e 4 f(t)γtdt by (2.7) = 0. by (2.6)](/img/revistas/ruma/v46n1/1a03128x.png)
So vanishes near
. However, since
is holomorphic, it follows that
. By Lemma 2.2,
![- t2- e 4 f(t)γt ≡ 0.](/img/revistas/ruma/v46n1/1a03133x.png)
Since and
are always positive,
![f ≡ 0.](/img/revistas/ruma/v46n1/1a03136x.png)
This shows that the functions in (2.5) form a complete basis of , and the lemma is proved. □
We combine Lemmas 2.1, 2.2 and 2.3 to show that:
Proposition 2.4 The function of (1.2) reproduces every element of
.
Proof: We claim that is dense in
. Identity (2.1) implies that
![∫ izt- t42 √ --n -z2 Rn e dt = (2 π) e .](/img/revistas/ruma/v46n1/1a03143x.png)
Applying repeatedly to both sides, we see that if
is a polynomial, then
![∫ izt - t24- - z2 2 Rn e p(t)e dt = p1(z)e ∈ A (TΩ),](/img/revistas/ruma/v46n1/1a03146x.png)
for another polynomial . Namely, the isometry in Lemma 2.2 sends every
![p(t)e- t24-∈ P e- t24 ⊂ L2(Rn, γ ) t](/img/revistas/ruma/v46n1/1a03148x.png)
to some
![-z2 -z2 2 p1(z)e ∈ P e ⊂ A (TΩ).](/img/revistas/ruma/v46n1/1a03149x.png)
By Lemma 2.3, is dense in
. Hence
is dense in
as claimed.
By Lemma 2.1, reproduces every element of
. Therefore, since
is dense in
, it follows that
reproduces every element of
. □
With this result, condition (ii) of stated in the Introduction is verified. Therefore, to prove the theorem, it remains only to check condition (iii). This is done by the following proposition.
Proposition 2.5 For each fixed
![K( - ,w) ∈ A2(T ). Ω](/img/revistas/ruma/v46n1/1a03163x.png)
Proof: Fix , and consider
. For any fixed
, formula (1.2) satisfies
![K Ω(z,w) = K Ω+ξ(z + iξ,w + iξ)](/img/revistas/ruma/v46n1/1a03167x.png)
for all . Further,
and
. Therefore, without loss of generality, we may assume that
in the statement of this proposition. We want to show that
. But
![1 ∫ eizt K(z, 0) = -----n ----dt, (2π) Rn γt](/img/revistas/ruma/v46n1/1a03173x.png)
and with Lemma 2.2, we get
![∥K(z, 0)∥2L2(TΩ) = (21π)n∥γ1∥2L2(Rn,γt) 1 ∫ t 1 = (2π)n- Rnγtdt = K(0, 0) < ∞, as 0 ∈ TΩ.](/img/revistas/ruma/v46n1/1a03174x.png)
We have shown that, for each ,
.
This result verifies condition (iii) of the Introduction, and the theorem follows.
References
[1] T. G. Genchev, Integral representations for functions holomorphic in tube domains, C. R. Acad. Bulgare Sci. 37 (1984), 717-720. [ Links ]
[2] T. G. Genchev, Paley-Wiener type theorems for functions in Bergman spaces over tube domains, J. Math. Anal. and Appl. 118 (1986), 496-501. [ Links ]
[3] I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products, Academic Press, (1980). [ Links ]
[4] S. Krantz, Function Theory of Several Complex Variables, 2ed. Wadsworth & Brooks/Cole, Pacific Grove 1992. [ Links ]
Ching-I Hsin
Division of Natural Science,
Minghsin University of Science and Technology.
Hsinchu County 304, Taiwan.
hsin@must.edu.tw
Recibido: 6 de mayo de 2004
Aceptado: 28 de marzo de 2005