Construction of szeg˝o and Poisson kernels in convex domains Текст научной статьи по специальности «Математика»

YflK 517.55

Construction of Szego and Poisson Kernels in Convex Domains

Simona G. Myslivets*

*

Institute of Mathematics and Computer Science Siberian Federal University Svobodny, 79, Krasnoyarsk, 660041

Russia

Received 26.09.2018, received in revised form 05.10.2018, accepted 06.10.2018

In this paper, we construct Szego and Poisson kernels in convex domains in Cn and study their properties.

Keywords: convex domain, Szego and Poisson kernels. DOI: 10.17516/1997-1397-2018-11-6-792-795.

This paper contains some results related to the construction of Szego and Poisson kernels in convex domains which are of significant importance for integral representations in such domains.

1. Construction the Szego kernel

Let D be a bounded domain in C" with a smooth boundary. Let H(D) be the space of holomorphic functions in D with the topology of uniform convergence on compact subsets of D, and H(D) be the space of holomorphic functions in a neighborhood of D with the corresponding topology. The space H(D) is the subspace in C2(dD) with respect to the measure dn on dD, where dn = g(Z)da, g(Z) € Cl(dD), g(Z) > 0, and da is the Lebesgue measure on dD. By the Maximum Modulus Theorem the mapping H(D) —> L2(dD) is injective. By H2 = H2(dD) we denote the closure of H(D) in L2.

Consider a restriction mapping r : H(D) —> H(D). The mapping r continues until continuous from H2 in H(D).

Lemma 1 (Lemma 4.1. [1]). The restriction mapping r : H(D) —>• H(D) is continuous, if H(D) is considered with topology induced by the space L2.

Therefore, the mapping r continues until a continuous map i : H2 —> H(D). In this case, we say that for functions f € H2 there is a holomorphic continuation f = i(f) in D. Further, this continuation will be denoted by the same symbol f.

In [1] ther was considered the Lebesgue measure da on the boundary of the domain, in our case for the measure dn = g(Z)da the proof is similar.

Since the space H2 is a Hilbert separable space, there exists an orthonormal basis

(1)

TO

f (z ) = Y. ck ^ (z)

(2)

k=i

* [email protected] © Siberian Federal University. All rights reserved

with respect to the basis (1), which converges in the topology of L2, where ck = (f, <k) = = f f (u)<pk(u) d^(u). Then

dD

f {C HE f(u)LPk(u) d^(u)¥k to) = f (u)Y] <pk (u)MC ) d^(u). fc=AJdD / JdD k=i

w

Denote K(Z,u) = ^^ <k (Z)<pk (u) and K(Z,u) G H(D) on Z G D for a fixed u G D.

k=l

Lemma 2. We can choose an orthonormal basis {<k }W=i in H2, which consists of functions <k in H(D).

Lemma 3. If D is a bounded strictly convex domain with a smooth boundary, then we can choose a polynomials basis {<k}W=i.

Further on, we assume that the basis is chosen in accordance with Theorem 5.1 [1]. According to this theorem the continuation of the kernel K(Z, u) has the property:

i(f )(z)= f f (Z)K(z,p) d^(Z), z G D,

■JdD

w

where K(z,p) = ^^ i(<k)(z)i(<pk)(Z) and the series converges uniformly on compact subsets of

k=l

D x D. This kernel we call the Szego kernel. Then

f (z)= I f (Z)K(z,p) d^(Z), (3)

■JdD

where f (z) is identified with f(z) = i(f )(z) and f G H2. We define the Poisson kernel

K(z,p) ■ K(Z, P) = K(z,p) ■ K(z,p) = IK(z,P)l2 ( ) K(z,P) K(z,P) K(z,P) '

w w

and K(z, z) = ^2 Pk(z)<pk(z) = E I<Pk(z)12 > 0. k=1 k=1

Lemma 4. The kernel K(z, z) > 0 for any z G D.

Lemma 5. A function f G H(D) satisfies the integral representation

f (z)= I f (Z)P(z,Z) dp(Z), (4)

JdD

for z G D.

Corollary 1. If the space H(D) is dense in the space H(D) nC(dD) = A(D), then a function f G A(D) satisfies the integral representation (4).

Suppose that the domain D satisfies the condition (A): for any point Z G dD and any neighborhood U(Z) the Szego kernel K(z, p) is uniformly bounded in z G D and z / U(Z). Further on, we assume that the domain D satisfies the condition (A).

Theorem 1. Let D be a strictly convex domain in Cn and the kernel K(z, p) satisfies the Holder condition with exponent ^ < a ^ 1 for Z G dD and a fixed z G D. Then the domain D and the kernel K(z,p) satisfy the condition (A).

Consider the restriction of the form

L(z,Z, Z)

£~i 4 dZ[k] a d(

K(Zi -zi) + ••• + P'zn(Cn - zn)]n to dD, it is

L(z,C, Z) =

K (Ci - zi ) + ... + Pi (Zn - zn)]n g(Z ) [p'(i (Zi - zi) + ... + p'Cn (Zn - zn)]n

_Z) dp(Z)_ J, Z , ,

= j-^-—- ' , -TTn = L(z, Z, Z) dp(Z).

[Pc1 (Z1 - z1 H ... + Pçn (Zn - zn)\

The proof of Theorem 1 shows that

K(z, Z) = L(z, Z, Z) (5)

for Z G dD.

Lemma 6. The function K (z, Z ) is unbounded as z ^ Z and Z G dD, z G D.

2. The Poisson kernel and its properties

For a function f G C(dD) we define the Poisson integral:

P[f](z) = F(z)= f f (Z)P(z,Z) dp(Z).

JdD

In strictly convex domain that satisfy the condition (A), from Equality (5) and the form of the kernel P (z, Z ), it follows that this kernel is a continuous function for z G D and then the function F(z) is continuous in D.

Theorem 2. Let D be a bounded strictly convex domain in Cn satisfying the condition (A), and f G C(dD), then the function F(z) continuously extends onto D and F(z)\dD = f (z).

Consider the differential form

n

w = (-1)k-1Zk dZ[k] A dZ, k=i

(n - 1)!

where c = —--—. Find the restriction of this form to dD for the domain D of the form

(2ni)n

D = {z G Cn : p(z) < 0}. Then by Lemma 3.5 [5], we get

dZk] A dZ = (-1)k-12n-1inda

dZk \ grad p\' Therefore, the restriction of w to dD is equal to

\ (n - 1)! ^^ z dp da

= w\dD = dZ 'ûradPï■

We denote

ít\ (n - 1)! ^ K dp 3(ZK dz

2nn k=i dZk \ grad p\'

1

Proposition 1. If D is a strictly convex circular domain, then g(Z) is a real-valued function that does not vanish on dD.

Therefore, we can assume that g(Z) > 0 on dD. Therefore, dn = gda is a measure and for it all previous constructions are true.

Proposition 2. Let D be a strictly convex (p1,... ,pn)-circular domain, i.e.

is real-valued and not zero.

References

[1] L.Bungart, Boundary kernel functions for domains on complex manifolds, Pac. J. of Math.., 14(1964, no. 4, 1151-1164.

[2] L.Hormander, An Introduction to Complex Analysis in Several Variables, North-Holland, Amsterdam, New York, 1989.

[3] L.A.Aizenberg, A.P.Yuzhakov, Integral Representations and Residues in Multidimensional Complex Analysis, Trans. Mathematical Monographs, American Mathematical Society, Providence, RI, 1983.

[4] E.M.Stein, G.Weiss, Introduction to Fourier Analysis on Euclidian Spaces, Princeton Univ. Press, Princeton, 1975.

[5] A.M.Kytmanov, The Bochner-Martinelli Integral and Its Applications Birkhauser, Basel, Boston, Berlin, Science, 1995.

Построение ядер Сегё и Пуассона в выпуклых областях

В этой статье строятся ядра Сеге и Пуассона в выпуклых областях пространства С" и изучаются их свойства.

Ключевые слова: выпуклые области, ядра Сегё и Пуассона.

p(Cu ■■■Xn)= p(Cieipie,..., Cueipn9 ), 0 < в < 2п,

where p1,... ,pn are positive rational numbers. Then the function

Симона Г. Мысливец

Институт математики и фундаментальной информатики Сибирский федеральный университет Свободный, 79, Красноярск, 660041

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