Fine-analytic functions in Cn Текст научной статьи по специальности «Математика»

УДК 517.9

Fine-analytic Functions in Cn

Azimbai Sadullaev*

Department of Mathematics National university of Uzbekistan University, 4, Tashkent, 100174 Uzbekistan

Received 07.09.2018, received in revised form 09.11.2018, accepted 13.04.2019 In this paper we study class of fine-analytic functions in the multidimensional space Cn. The definition of fine-analytic functions in the multidimensional case differs somewhat from the well-known definition of fine-analytic functions on the plane. We give a relationship between classical notion of fine-analyticity and fine-analyticity in Cn.

Keywords: Gonchar class, finite order functions, rational approximation, fine-analytic functions, pluripo-lar sets.

DOI: 10.17516/1997-1397-2019-12-4-444-448.

1. Introduction and preliminaries

In [17] (see also [16]), we established the following connection between Gonchar class functions with fine-analytic functions in complex space Cn:

Theorem 1.1 (A. Sadullaev, Z. Ibragimov). Let K C Cn is a nonpluripolar set and f € C (K) is a continuous funstion on it. If f belong to the Gonchar class RK of finite order, i.e., there exist a sequence of rational functions

rm (z) = Pm (z), deg rm < m, m = 1, 2,...,

qm (z)''

such that

\\rm -

<

1

г1/« '

1, 2,..s < ж,

then f fine-analytically continues to the whole space Cn. That is, there is a fine-analytic function f on Cn, such that f\K = f.

The definition of fine-analytic, more specifically (W2)fine-analytic functions (see Definition 3) in the multidimensional case differs somewhat from the well-known definition of fine-analytic functions on the plane. In this paper we give comparisons of these definitions in the one-dimensional case, we give examples and indicate the difficulties in determining fine-analytic functions in Cn by the standard way.

We first recall the definition of fine-analytic functions. They are determined by means of fine(thin) topology. A fine topology in Cn is the weakest topology in which all plurisub-harmonic (psh) functions are continuous. A fine topology is generated by sets of the form {u (z) < a} , {u (z) > a} , u € psh (Cn). Fine neighborhood of a point a is an open set

* sadullaev@mail.ru © Siberian Federal University. All rights reserved

m

V C Cn, a € V, in the fine topology for which the complement W = Cn\V is thin at the point a, i.e. there exists a ball B = B (a, r) and a plurisubharmonic in B function u € psh (B) : lim u (z) < u (a). A closed fine neighborhood of a point is a compact of

z^a,zEW

the form B (a, r) \G, where r > 0, G C B (a,r) is an open, thin set at the point a. In the general case, in a literature, a fine neighborhood is any set V C Cn, a € V, for which Cn\V is thin at the point a (see, for example, [2]).

Definition 1. A function f (z) is called fine-analytic in a planar domain D € C if

1) it is defined almost everywhere with respect to the capacity in D i.e., outside of some polar set E C D it admits a finite value;

2) for each point a € D\E there is a closed fine neighborhood F of a such that f € R (F), or, equivalently, the restriction f |f is uniformly approximated by rational functions on F.

The notion of fine-analyticity was introduced and used by B.Fuglede [8-10] for a class of functions that have the Mergelyan property in fine neighborhoods of a point. If a function f(z) is analytic in a neighborhood of a point a, then f has the Mergelyan property in arbitrary compact K 9 a, i.e f is uniformly approximated on K by rational functions. Fine analyticity of f in the point a means uniformly approximation by rational functions only on some fine neighborhood K 9 a. In work of A. Edigarian and J.Wiegerinck [3,4], A. Edigarian, S.El Marzguioui and J.Wiegerinck [5], S.El Marzguioui and J.Wiegerinck [14], J.Wiegerinck [18], T.Edlund [6], T.Edlund and B.Joricke [7] fine-analytic functions were studied for their applications in pluripotential theory, more precisely, in the description of pluripolar hulls, in the establishment of pluripolar hulls of graphs r = {w = f (z)}. On pluripolar hulls of analytic sets and graphs r = {w = f (z)} see also the papers of N. Levenberg, G. Martin and E. Poletsky [12], N. Levenberg and E. Poletsky [13].

In [15] the author constructed a function f € O (U) p| C(U), where U is disk, such that the pluripolar hull rf = rf. It is clear that if f (z) holomorphically extended to some point z0 € dU, then the point (z°,f (z0)) € rf. It is natural to expect the opposite, that if (z°,f (z0)) € rf, then f (z) will be holomorphic at the point z°. But this assumption was refuted by A.Edigarian and J.Wiegerinck [4], who constructed a function f, that is holomorphic only inside the unit disk, for which rf = rf. J.Siciak, studying this example of Edigarian and Wiegerinck, established that the function f is analytically "pseudo-continued" through boundary points dU and the pluripolar hull rf always contains a graph of the pseudo-continued function; however, the condition of pseudo-continuity is not necessary for rf = rf. In [7] T.Edlund and B. Joricke showed that a fine-analytic continuation is well suited for describing the pluripolar hull of the graphs.

Unfortunately, the Definition 1 has one drawback, that the elementary functions, such as

ft ^ iexP~, for z = 0 m

f (z) = < z (1)

0, for z = 0

is not fine-analytical, although outside the point z = 0 it is represented as

1 1

f (z) = 1 + z + 2!z + ...

and belong to the Gonchar class R. In connection with this example, we will extend the class of finely analytic functions by slightly weakening the condition.

Definition 2. A function f (z) is called (Wl)fine-analytic in a domain D € C if

1) it is defined almost everywhere with respect to the capacity in D i.e., outside of some polar set E C D it admits a finite value;

2) for each point a € D\E there is a closed fine neighborhood F of a,such that there exists a sequence of rational functions rm (z), m = 1, 2, 3 ..., with poles outside F : rm (z0) ^ f (z0) as m ^ m Vz0 € F. Moreover, rm (z) converges uniformly to f (z) inside of the set F\ {a} , in the sense that for any compact F' C K\ {a} , \\rm — f ||F, ^ 0 as m ^ m.

As mentioned in the work of Bedford-Taylor [1], in contrast to the planar case, in the multidimensional space Cn, n > 1, the notion of thin set (i.e., a set, that is thin in any of its point) does not coincide with the concept of a pluripolar set. For these and for the complexity of the structures of R (F) in the multidimensional space Cn we should point out that in Cn, n > 1, we could not give definition of fine-analytic functions as in the Definition 1. For example, even the rational function

r (zuz2)Jzi/z2 if (z!,z2) = (0,0), (2)

\0 if (z-i,z2) = (0,0)

is not fine-analytic in the sense of Definition 1, since the pluripolar set {z2 = 0} is not thin at the point (0,0) € C2. It is convenient for us to define fine-analytic functions in Cn as follows.

Definition 3. A function f (z) is called (W2)fine-analytic in a domain D € Cn if there is an increasing sequence of close sets Fj C Fj+i C D, j = 1, 2,..., such that:

1) the condenser capacity C (B\Fj, B) ^ 0 as j ^ m for each ball B CC D. It follows, that the set D\ Fj is pluripolar, but the convers is not true;

js

2) f admits a finite value everywhere in |J Fj;

j

3) For each ball B CC D and for each number j, the restriction fBp| Fj can be uniformly approximated by rational functions on B p| Fj, i.e. f\b q Fj € R (F), j = 1, 2,....

We note that the function 1

I exp - for z = 0,

f(z) = < z

0 for z = 0

given in (1), which is not fine-analytic in the sense of Definition 1, is (W1) and (W2) fine-analytic

on the plane C: at the point a = 0 we can put F = {\z\ < 1}, and the sequences of rational

functions we construct in the following way. We set

„ r, , 1 n 1 n 1]

Fm = {\z\ < 1}\{ 0 < \z\ <—, —- — — < 'Arg z<- + - .

[ m 2 m 2 m j

Then f € R (Fm) and there exists a rational function rm (z) : \\rm — f ||F < —. It is easy

Fm m

to see that the sequence {rm(z), m = 1,2,... } satisfies the conditions of Definitions 2 and 3.

2. Relationship between definitions 1-3 in C

Let a function f (z) is fine-analytic in a domain D C C in the sense of Definition 1, i.e. it admits a finite value outside of some polar set E C D and for each point a € D \ E there is a closed thin neighborhood F B a : f\F € R(F). Then D \ E is finely open set, in which the function f (z) is fine-analytic in the sense of B.Fuglede [8] (see, also [5,10]). It is clear that from Defition 1 follows Definition 2, Def 1 ^ Def 2. As an example of the function (1) shows, the converse is not true. We prove the following theorem.

Theorem 2.1. From Defition 3 follows Definition 2, Def 3 ^Def 2.

Proof. Let f (z) be (W2)fine-analytic in D, i.e. there is an increasing sequence of closed sets Fj c Fj+i c D, j = 1, 2,..., such that, the condenser capacity C (B\Fj, B) — 0 as j — x> for each ball B cc D and the restriction f^ F. e R (F), j =1, 2,... .

We take a e |J Fj, assuming without loss of generality, a e Fl. By assumption of Definition 3

j

there exists a sequence of rational functions rj (z) : \\rj — f < -. We can assume that f (a) =

j j 1

= rj (a) = 0. Then for each j e N there exists a ball Bj = B (a,£j) CC D : \\rj \\B, < -, £j > 0.

j j

We can assume, that el > £2 > ..., and £j — 0 as j ^

Now we use the following Wiener criterion (see [2,11]): an open set E c C is thin in a point z0 € dE if and only if

f CPip<

J o P

where C(p) is the capasity of E n B(z0,p).

Fix a ball B = B (a,£) cc D, £ > £l. Since C (B\Fm) 0 as m ^ <x, then there exists

an increasing sequence of natural numbers mj : C (B\Fmj) < £j+l. We put Kj = Bjf] Fmj

oo

and K = |J Kj. Then K is compact and a e K. For any £j+l < p < £j we have C(p) =

j=i

C (B(z0,p)\K) < C (B\Fmj) < £j+i < p. It follows, that

i1 CP ip < 1

Jo P

and by Wiener criterion B\K is thin at the point a e K.

From rj (a) = f (a) = 0 it follows that rj (a) — f (a). In addition, if a compact F' c K\ {a} , then F' c Fj0 for some j0. Therefore, \\rj — f \\F, — 0 as j — x>. Theorem is proved. □

Problem. We do not know if Defition 3 will follows from Defition 1 or 2.

This work was received during my visit to ICTP (International Centre for Theoretical Physics), 17.08-17.09.18. I would like to thank the head of the ICTP for invitation and for creating an excellent condition of stay.

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Тонко-аналитические функции в C

Азимбай Садуллаев

Department of Mathematics Национальный университет Узбекистана Университетская, 4, Ташкент, 100174

Узбекистан

В 'работе исследуются тонко-аналитические функции в многомерном комплексном пространстве С". Определение тонко-аналитических функций в многомерном случае п > 1 несколько отличается от плоского случая п = 1. Мы сравниваем эти определения, в примерах показываем существенные их различия и необходимость использования именно предлагаемого в работе определения при п > 1.

Ключевые слова: класс Гончара, функции конечного порядка, рациональная аппроксимация, тонкоаналитические функции, плюриполярные множества.