On holomorphic continuation of integrable functions along finite families of complex lines in an n-circular domain Текст научной статьи по специальности «Математика»

УДК 517.55

On Holomorphic Continuation of Integrable Functions Along Finite Families of Complex Lines in an n-circular Domain

Bayram P. Otemuratov*

Karakalpak State University Nukus, 230112 Uzbekistan

Received 06.07.2017, received in revised form 16.08.2017, accepted 30.11.2017 This paper contains some results related to holomorphic extension of integrable functions defined on the boundary of D С Cn, n > 1 into this domain. We shall consider integrable functions with the property of holomorphic extension along complex lines. In the complex plane C the results about functions with such property are trivial. Therefore, our results are essentially multidimensional.

Keywords: integrable functions, holomorphic extension, Szego kernel, Poisson kernel, complex lines. DOI: 10.17516/1997-1397-2018-11-1-91-96.

Earlier, in papers [1-3], there were considered sufficient conditions for holomorphic extension of integrable functions for families of complex lines passing through an open set in D, through a germ of generating manifold in a complex hypersurface.

The example of Globevnik [4] shows that for fnctions continuous on a sphere a family of lines passing through n points is not sufficient for holomorphic continuation. In the paper [5] there were considered families of complex lines passing through n +1 points lying in an n-circular domain D in Cn and functions continuous on its boundary. In this paper we generalize this result for integrable functions.

Let D be a complete strictly convex bounded domain in Cn with smooth boundary centered at the origin, i.e., together with each point z0 = (z0,..., zn) £ D it contains the polydisc

{z G Cn : \zk\ < \z0k\, k = 1,

г}.

Denote by D+ = ..., \zn\) ■ z € D} the image of D in absolute orthant

R+ = {(X1, Xn) ■ Xk > 0, k = 1,. .. ,n}.

Let dD+ = {(\zi\,..., \zn\) ■ z € dD}.

Consider a finite measure n on dD+. The measure n is called massive on the Shilov boundary [6, Sec. 11] if for any subset E C dD+ of zero measure (with respect to n) the following condition holds: dD+ \ E D S(D+), where S(D+) is the image of the Shilov boundary S(D) in the absolute orthant. In our case S(D+) = dD+. It follows from Theorem 3.1 [7] that the Lebesgue measure ^ on the boundary of such domain is massive. Henceforth we shall always assume that the measure H is massive.

Define the Szego kernel of the domain D

a>k

a a

aZ z

(1)

where

1

1

/ \z\2adM / \Ci\2ai •... • \Zn\2andM dD+ dD+

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

a

a

and a = {ai,...,an} is a multi-index such that a ^ 0 (i.e. ak ^ 0, k = l,...,n), and za = z?1 ■ ... ■ , \\a\\ = ai + ... + an.

Recall the definition of the class Hp(D): a holomorphic function f belongs to Hp(D) (p > 0)

if

sup í \f (Z - ev(Z))\pda<

£>0 JdD

where da is a volume element of the surface dD, and v(Z) is the unit vector of the external normal to the surface dD at the point Z. It is well known that the normal boundary values of a function f e Hp(D) belong to the class Lp(dD) (with respect to the measure da).

The existence of Szego kernels for n-circular domains is given by the following theorem:

Theorem 1. Let f be a finite measure on dD+. For any function f e Hp(D), (p ^ 1), there is a Szego integral representation

f (z) = lim -i— i dff f (Z)h(Z, rz)z e D, (2)

r^i (2m)n J9D+ JA{(1 Z

where

AK\= {Z : Zi = IZi \eiSl ,...,Zn = \Zn\ei°n, 0 < ek < 2n, k =1,...,n, \Z \e dD+},

dZ = dZi dZn Z = Zi ■ Zn '

and the Szego kernel h(Z, z) = h(Z1z1,..., Znzn) is in O(D) with respect to Z for fixed z e D and in O(D) with respect to z for fixed Z e dD if and only if the measure f is massive.

For continuous functions the theorem is proved in [6], for functions of the class Hp it can be obtained by approximating a function f (z) by functions f (rZ) as r ^ 1,r < 1, in the metric of Hp. Thus, by theorem 1 the series (1) converges absolutely for Z e D and z e D and uniformly for Z e D and z e K, where K is an arbitrary compact subset of D.

It is clear that dD = |J A|z|. Let us point out an obvious property of the Szego kernel: |C|edD+

h(Z,z) = h(Z, Z) = h(z,Z).

Introduce the Poisson kernel

P = h(Z,z)h(Z,Z) = \h(Z,z)\2 .

' h(z, z) h(z, z)

Note that the kernel P(Z, z) is defined for Z G D and z G D since h(z, z) > 0. Proposition 1. If f G Hp(D) (p > 1), then

f (z) = lim -i— i dpi f (Z )P (Z, rz) z G D.

r^1 (2ni)n JdD+ JAK| C

Proof. Using formula (2), we get

1 f J f stents d _ 1 f , f .f^h(Z,z)h(C,Z) dZ

¿wL(Z)PtZ'z>7 = (ànL(Z>! Z

1 1 P .1 P d 1

№ H-z,z)JdD+ d* L f (0HZrZ)ï hZz)Z = W^)f (z)h(zZ = f (z),

since the function f (Z)h(Z, z) is holomorphic in Z £ D for fixed 2 £ D. □

By Lemma 1 from [5] for Z = z:

h(Z,z) = J2 a,a\z\2a > 0

in D, and h(z, z) ^ to as z ^ dD.

Assume that the domain D satisfies the condition (A): h(Z, rz) is uniformly bounded in z outside any neighborhood of Z if Z,z £ dD and Z = z, as r ^ 1.

Theorem 2. If a domain D satisfies the condition (A) and f £ Lp(dD), then the Poisson integral

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

T^I (2rn)n JdD+ JA{(1 Z

gives a real-analytic function in D, and its boundary values in the metric of Lp coincide with values of f on the dD.

Proof. Real analyticity of F(z) follows from the real analyticity of the Szego and Poisson kernels. From condition (A) and Lemma 1 [5] it follows that P(Z, rz) tends uniformly to zero outside of any neighborhood of the point Z for Z, z £ dD, Z = z and r ^ 1. Moreover, P(Z, z) > 0 and P[1](Z) = 1. Hence, the Poisson kernel P(Z,z) is an approximative unit [8, Th. 1.9]. □

Here we use the notation dZ = dZi A ... A dZn, d([k] = d(1 A ... A dQh-1 A dQh+1 A ... A d(n. The denominator of the kernel p^ (Z1 — z1) + ... + p'^ (Zn — zn) = 0 for Z,z £ dD and Z = z. Indeed, the equation p^ (Z1 — z1) + ... + p'^ (Zn — zn) = 0 determines the complex tangent plane to dD at the point Z. If D is strictly convex then the tangent plane intersects the boundary of the domain only at the point Z. The Szego kernel of D is expressed by the Leray kernel according to Corollary 26.13 [6] and the form of its denominator does not vary, therefore such domains satisfy the condition (A).

Consider a differential form

w = c^(-1)k-1ä dÇ[k] dZ, k=i

(n — 1)!

where c = —--—. Find the restriction of this form of narrowing on dD for a domain of the

(2ni)n

form

D = {z £ Cn : p( |zi|2,..., \zn\2) < 0},

where p(z) is twice differentiable function, and gradp = (..., =0 on dD.

\dz1 dznJ

Denote \zk\2 = tk, k = 1,... ,n. Then

( dp dp \ grad p = \at1 zi,...,dï~ zn) =0.

The function p can be chosen such that \ gradp\\dD = 1. Let v = ^\dD, then, as can be easily verified (see, e.g., [9, Lemma 3.5]),

n dp n dp

v = c> Zh—=- da = c> th —— da,

^ dZh ^ h dth '

h=1 Sh h = 1 h

where da is the Lebesgue measure on dD. In the case of n-circular domains da = da+ ■ da'

where da' is the measure defined by the differential form

1 dZl A A dZn

(2ni)n Zi Zn '

and da+ is the Lebesgue measure on dD+. Therefore

n dp

v = cS^tk -d— da+ ■ da'. t=i dtk

We denote

n dp

M = cJ2tk if da+ . (3)

k=i dTk

Lemma 1. If D is a complete n-circular domain, then p is a measure on the dD+.

The proof of the lemma given in [5].

Corollary 1. If D is a complete n-circular strictly convex domain, then p is a massive measure on dD+.

We consider a modified Poisson kernel

h(C,z)h(C,w)

h(w, z)

Then if w — Z we get Q(Z, z, z) — P(Z, z), and h(z, z) > 0. Therefore, there exists a neighborhood U of the diagonal w = Z in Dz x Dw, wherein h(w, z) = 0. Consider the function

Hz,w) = cf f(Z)Q(Z,z,w)dv = •I dD

' dD

ci dp f f (Z)Q(Z,z,w)dZ, (z,w) G D x D. J3D+ JA Z

This function is holomorphic in (z, w) G U, and for w = z we have &(z, w) = F(z) and

F (z)

dz5 dwY

where

=2 dz5 dzY ' dS+Y z,w) d Si+.-.+Sn+n+.-.-yn z,w)

(4)

dz5dwY dzS1 ■■■ dztdwYi ■■■ chWf dS+Y F ( z) d 5i+---+5n+Yi+---Yn F (z)

dzs dzY dz J1 ■■■ dzt dzYl ■■■ dzf

and S = (Si,..., Sn), Y = (yi, .. .,Yn).

Let Z = bt, b G CPn~ 1, t G C, then, as shown in [10] (see also [9, Sec. 15])

w = d A X(b), (5)

where X(b) is a differential (n — 1,n — 1)-form that does not depend on the t.

In what follows we shall assume that there is a direction b0 = 0 such that

{b0, 0 =0 для всех С e D. (6)

Denote by a set of complex lines of the form

Izb = {С e Cn : Cj = Zj + bj t, j = i,...,n, t e C}, (7)

passing through the point z e Г in the direction of the vector b e CPn-1 (the direction b is defined up to multiplication by a complex number Л = 0).

By Sard's theorem for almost all z e Cn and for a fixed be CPn-1 the intersection lz ь П dD is a finite set of of piecewise smooth curves (except for the degenerate case where dD П lz,b = 0).

It is known that if f e Lp(dD),p > 1, then for almost all z e D and almost all b e CPn-1 the function f e Lp(dD П lz, b) (see [1]).

We say that the function f e Lp(dD) possesses one-dimensional holomorphic extension property along complex lines lz,b e of the form (7), if for almost all lines lz,b such that dDnlz,b = 0 there exists a function f with the following properties

1) f e Hp(D n lz,b),

2) normal boundary values of f in the metric of Hp coincide with values of f on the set dD n lz,b almost everywhere.

Consider the the Bochner-Martinelli kernel

U (Cz> = n-F Ё f-1)'-1 СФ d°WA dc,

where dC = dC1 A ... Л dCn, and dC[k] is obtained from dC by omitting the differential dCk. For a function f e Lp(dD) we define the Bochner-Martinelli integral

F(z)= f f (C)U(C,z), z edD. (8)

■JdD z

The function F(z) is harmonic outside the boundary of the domain and converges to zero as \z\ ^ ж.

We call the set Lr to be sufficient for holomorphic continuation, if the fact that f e Lp(dD) has the one-dimensional holomorphic extension property along almost all complex lines of the family Lr implies that the function f extends holomorphically into D as a function of the class Hp.

Theorem 3. Let D be a bounded n-circular strictly convex domain and function f e Lp(dD) has the one-dimensional holomorphic extension property along complex lines passing through

ds §(z,w)

the origin, then Ф(0, w) = const and -d^Y

than ||£||.

is a polynomial in w of degree not higher

z=0

For continuous functions Theorem 3 has been proved in [5].

References

[1] B.P.Otemuratov, On functions of class Lp with a property of one-dimensional holomorphic extension, Vestnik KrasGU. Ser. fiz. mat. nauki, Krasnoyarsk, (2006), no. 9, 95-100 (in Russian).

[2] B.P.Otemuratov, On multidimensional theorems of Morera for integrable functions, Uzb. mat. zh., Tashkent, (2009), no. 2, 112-119 (in Russian).

[3] B.P.Otemuratov, Some sets of complex lines of minimal dimension sufficient for holomorphic continuation of integrable functions, Journal SFU. Mathematics and physics. Krasnoyarsk, 5(2012), no. 1, 97-106 (in Russian).

[4] J.Globevnik, Small families of complex lines for testing holomorphic extendibility, Amer. J. of Math., (2012), no 6, 1473-1490.

[5] A.M.Kytmanov, S.G.Myslivets, Holomorphic extension of functions along finite families of complex straight lines in an n-circular domain, Sib. Math. J., 57(2016), no. 4, 618-631.

[6] L.A.Aizenberg, A.P.Yuzhakov, Integral representations and residues in multidimensional complex analysis, Novosibirsk, Nauka, 1979 (in Russian).

[7] G.M.Henkin, Method of integral representations in complex analysis, Results of science technology. Modern problems of mathematics., vol. 7, Moscow, 1985 (in Russian).

[8] I.Stein, G.Weis, Introduction to Fourier analysis on Euclidean spaces, Moscow, Mir, 1974 (in Russian).

[9] A.M.Kytmanov, Bochner-Martinelli integral and its applications, Novosibirsk, Nauka, 1992 (in Russian).

[10] A.M.Kytmanov, S.G.Myslivets, Higher-dimensional boundary analogs of the Morera theorem in problems of analytic continuation of functions, J. Math. Sci., 120(2004), no. 6, 1842-1867.

О голоморфном продолжении интегрируемых функций вдоль конечных семейств комплексных прямых в n-круговой области

Байрам П. Отемуратов

Каракалпакский государственный университет Ч.Абдирова, 1, Нукус, 230112 Узбекистан

Статья содержит результаты, связанные с голоморфным продолжением интегрируемых функций, заданных на границе области D С Cn, n > 1, в эту область. Речь идет об интегрируемых, функциях с одномерным свойством голоморфного продолжения вдоль комплексных прямых. На комплексной плоскости C результаты о функциях с одномерным свойством голоморфного продолжения тривиальны, поэтому наши результаты существенно многомерны.

Ключевые слова: интегрируемые функции, голоморфное продолжение, ядро Сеге, ядро Пуассона, комплексные прямые.