Analytic capacity and rational approximations

3.7.

ANALYTIC CAPACITY AND RATIONAL APPROXIMATIONS*

Let E be a bounded set in on ~ \ [

and bounded on ~

~

; B(E,

I) be the set of all functions f on ~

in modulus by unity, f(~) = 0; let ~C[,~)

set B(E, I), consisting of functions continuous on E we mean the number

~ .

By the

, analytic

be the subset of the

analytic capacity

T(E) of the set

~CE,O z~ By the

analytic C-capacity

~(E) of the set E we mean the number

The concept of analytic capacity has been introduced by Ahlfors [5] in connection with Painleve's problem regarding the description of the sets of removable singularities for the class of bounded analytic functions. Ahlfors [5] has shown that these sets are characterized by the fact that T(E) = 0. However, it is desirable to find the description of the sets of zero capacity in metric terms. Conjecture I. Let E be a compactum in ~ ; we have T(E) = 0 if and only if the projections of the set E on almost all (relative to the linear measure on the circumference) directions have zero linear measure (such sets are called irregular if their linear Hausdorff measure is positive). If the linear Hausdorff measure of the set E is finite and x(E) = 0, then the mean value of the measures of the projections is equal to zero. This follows from Calder6n's result [20] and from the known theorems regarding irregular sets (see [12, pp. 341-348]). The connection between capacity and measures is presented in detail in [34]. The capacity characteristics are particularly effective in the theory of approximations [6, 35, 7, 36]. A series of approximation problems is based on the unsolved problem regarding the semiadditivity of the analytic capacity:

~CEuF)~c[~CE)+~CF)], where c is an absolute constant and E and F are arbitrary disjoint compacta. Let A(K) be the algebra of all functions which are continuous on the compactum K, K C ~ and analytic at all the interior points of the set K; R(K) is the algebra of the uniform limits on K of rational functions of a complex variable with poles outside K; 8~ is the interior boundary of the set K, i.e., that part of the boundary of the compactum K which is not contained in the union of the boundaries of the components of its complement. The question that for which compacta do we have A(K) = R(K) has been solved in terms of the analytic capacity [6]. A derivation of the geometric conditions of approximability from these results requires an additional investigation of the properties of the capacity. Conjecture

2.

If ~(~~

= 0, then A(K) = R(K).

A positive answer to the question regarding the semiadditivity would mean the validity of Conjecture 2. Since the analytic C-capacity of a set of finite Hausdorff linear measure is equal to zero, this would give a proof for the following statement: Conjecture 3. If the Hausdorff linear measure of the interior boundary of the compactum K is finite, then A(K) = R(K).

*A. G. V I T U S H K I N a n d M . S. MEL'NIKOV. V. A. Steklov Mathematical Institute, Academy of Sciences of the USSR, UI. Vavilova 42, Moscow, 117333, USSR. Moscow State University, Department of Mathematics and Mechanics, Lenin Hills, Moscow, 117234, USSR.

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However, it is possible that the proof of Conjecture 3 can be obtained by evading the question regarding the semiadditivity of the capacity. One has not even proved that A(K) = R(K) if the Hausdorff linear measure of the interior boundary of K is equal to zero. The semiadditivity of the capacity has been proved only in some special cases ([19, 3740]), for example, for sets E and F, separated by a line. For a detailed discussion of this and of some related problems, see [9].

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