Carleman’s formula for a matrix polydisk Текст научной статьи по специальности «Математика»

УДК 517.55

Carleman's Formula for a Matrix Polydisk

Bahodir A. Shoimkhulov Jorabek T. Bozorov*

National University of Uzbekistan VUZ Gorodok, Tashkent, 100174

Uzbekistan

Received 20.01.2015, received in revised form 24.03.2015, accepted 06.05.2015 In this work an integral formula for a matrix polydisk is obtained. For a function from the Hardy class it allows to recover its value at any interior point from its values on a part of the Shilov boundary.

Keywords: Carleman's formula, matrix polydisc. DOI: 10.17516/1997-1397-2015-8-3-371-374

1. Statement of the problem and preliminaries

Consider a class of holomorphic functions in a domain D that behave reasonably well near the boundary 3D. Carleman's formulas solve the problem of recovery of a function from such a class from its values on a set of uniqueness M c dD for this class, which does not contain the Shilov boundary of D.

One-dimensional and multidimensional formulas were studied in the monograph [1]. In the paper [2] a new method is proposed for finding Carleman's formulas in homogeneous domains using a domain's automorphisms. In the present paper we use this method for a matrix polydisk.

Let Z = (Z1, Z2,..., Zn) £ Cn[m x m] be a vector of quadratic matrices of order m over the field of complex numbers C. The unit matrix disk is defined as the set

t = {Z £ C [m x m] : ZZ* < I} ,

where Z * = Z' is the conjugate transpose of the matrix Z, the notation ZZ * < I (I is the unit [m x m]-matrix) means that the Hermitian matrix I — ZZ * is positive definite. The skeleton of a matrix disk is the set

S (t ) = {Z £ C [m x m] : ZZ * = I} . The direct product of matrix disks

T = Tn = tn = {Z = (Z1,..., Zn) : Zj £ t, j = 1,..., n}

is called the matrix unit polydisk T = Tn in the space Cn [m x m].

The set S(T) = S(t) x • • • x S(t) is called the skeleton of T (see [3]).

To formulate the main result, we need the following Carleman formula for the unit matrix disk obtained in [2]. Let the set M c S(t) have positive measure ^1M > 0. Denote

M0,u = {£ : £ £ M, £ = Am, |A| = 1} , u £ SU(m), (1)

M'0 = {u : u £ SU(m), m1M0,u > 0} ,

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

where SU(m) is the group of special unitary matrices, i.e. det(u) = 1, m1 is the normed Lebesgue measure on the unit circle dU. Further on, let

= exp^0, where ^o(£) = tt— / •

2ni JMo,u n - A n

The Hardy class H 1(D) is defined as follows: a holomorphic in D function f (z) belongs to the class H 1(D) if

sup / |f (r£)| d^i < ro,

0<r<W S(D)

where is the Lebesgue measure on the skeleton S(D). Lemma 1 ( [2]). Lei f G H 1(t). Then the formula

f (0)

lim

/M, d^i i^œ

f (0

'M

^c(e) L^q(O)J

(2)

holds.

m

2. The main part

Let ) be an automorphism of a unit matrix polydisk Tn rearranging the points A G Tn

and 0. Such an automorphism has the form (see [4, p. 84])

)= (^(z1),..., $A(zn)) ,

where

- * - — 1 - — 1 (Zj) = Qj(Zj - Aj)(I - (Aj)*Zj) (Rj) 1,

Qj and Rj are [m x m]-matrices satisfying the following conditions

Qj (i - Aj AjQj' = I, Rj (i - Aj Rj' = I, Qj Aj + Aj Rj = 0.

In particular, for A = 0

$o(Z ) = ($0(z1),..., $"(z"))

and

$0(Zj) = Qj Zj (Rj)—1.

Let E = (E1,..., E") C S(T) and ^E > 0; = (£2,...,£n) G S (Tn—1). Define the sets

E0,? = {Z : Z G E, $1(Z1) = A, (zj) = A$0(£j)j = 2>, A G S(T)} ,

E0 = {Z G E : E0,? > 0} . Lemma 2. Lei f G H 1(Tn). Then the following formula

f(0) = ^-lim f f(Z)i^ 11

JEo «Mm(n—1) 7E

holds. Here ^>0 = exp and

^ (e) = 21- f

2ni Jei

0

where E^ is defined the same as in (1).

Mo)J

n + A dn

d^ (3)

e1 , n - A n

o.i1

Proof. Let x£ = (£2,..., £n) £ S(Tn-i). Then by Lemma 1 we have

f (0) = ,1™ / f (£)[fo(£)/fo(0)]'dMi.

Jeo,£ JEo.i

Integrating both parts of this equation over E0, we obtain formula (3) since

f(£)

i0,i

fo(£)

Lfo (0)J

<

s(t )

f(£)

fo(£)

Lfo(o)J

d^i

+

s(t )\eo,£

f(£)

fo(£)

Lfo(o)J

d^i

<

< If(0)| + / |f(£)| dMi.

J S(T)

Here we use the facts that ^0(Z) is the "quenching" function, and the Cauchy-Szego formula is applicable for the function f in t (see [4, p.93]). Therefore the transition to the limit under the integral sing is possible by the Lebesgue theorem. □

Formula (3) recovers the value of the function f at the point 0 from its values on E. Now, with the help of this formula, we shall prove a formula recovering the value of f at an arbitrary point of the domain Tn.

Denote

x£ = (£2, ...,£n) £ S(Tn-i), ea,5 = {Z : Z £ E, $A(Z1) = A, $A(£j) = A$A(£j), j = A £ s(t)} .

This set is measurable with respect to the measure m1 for almost all A and x£. Denote by Ea the set {x£ : x£ £ S(Tn-1), > 0}. The Fubini theorem implies that the (mn — m)-

dimensional Lebesgue measure of this set is positive. Denote

f A = exp ^A, -0A (£) = J 1

n + A dn

E11 1 n - A n '

A1 ,W1

where Eai,wi = {£1 £ E1,£ = ($A) 1 (a($a) v)) , |A| = l} , W £ $A(SU(m)).

Theorem 1. Let f £ H 1(Tn), E c S(Tn), m(E) > 0. Then for an arbitrary point A £ Tn we have the following formula

f (A)

Mmn-m($-1 (Ea)) l—™ JE

lim

f (Z )

fA(Z ) f a(A)

l n

,ZJJdM(Z ). (4)

j=i

Here

H (Aj ,Zj ) =

det(I(m) — Aj Zj )m is the Cauchy-Szego kernel for the (matrix) generalized disk.

Proof. For A = 0 the statement of Theorem 1 was proved in Lemma 2. Let fA(£) = f ($-1(£)). Then fA(0) = f (A). It is known that fA £ H1 (Tn) (see [2]). By applying Lemma 2 to fA and then performing the reverse transformation, we obtain

f (A)

SEa dmn-m WGEA)

lim

i—

f(£)

f a(£)

fA(A)

dm ($a(£)).

(5)

m

1

E

Since

dM (*a(0) = |det J(£,A)| dM(£), where J (A, £) is the Jacobi matrix of the transformation ), and (see [4])

n • -1 • —i

П H(Aj )h(ej )

det |J(£,A)| = ^-n--- ,

п H (Ai ,Aj)

jnl

n _-

the application of formula (5) to the function f (£) П H-1(£j, A) G H 1(Tn) gives (4). □

jnl

References

[1] L.A.Aizenberg, Carleman's Formulas in Complex Analysis. Theory and Applications, Kluwer Acad. Publ., Basel, Heidelberg, 1993.

[2] A.M.Kytmanov, T.N.Nikitina, Analogs of Carleman's formula for classical domains, Math. Notes, 45(1989), no. 3, 243-248.

[3] G.Khudayberganov, A.M.Kytmanov, B.A.Shoimkhulov, Complex Analysis in Matrix Domains, SFU, Krasnoyarsk, 2011 (in Russian).

[4] Hua Lo-Keng, Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains, American Mathematical Soc., RI, 1963.

Формула Карлемана для матричного полидиска

Баходыр А. Шоимкулов Жаробек Т. Бозоров

В 'работе получена интегральная формула для матричного полидиска. Для функций из класса Харди дано утверждение, позволяющее восстановить значения функции во внутренних точках полидиска по ее значениям на части границы Шилова.

Ключевые слова: формула Карлемана, матричный полидиск.