This paper marks one more important progress in the study of the properties of removable sets for bounded analytic functions. The continuous analytic capacity of a compact set is defined as , where the supremum is taken over all functions continuous in , analytic in and satisfying . For a general , one defines , compact}. This quantity was introduced by Erokhin and Vitushkin in the 1950’s in the context of the theory of uniform approximation by rational functions [see A. G. Vitushkin, Russ. Math. Surv. 22, 139–200 (1967; Zbl 0164.37701)].
In the present paper, the author proves that the continuous analytic capacity is semi-additive:
where are Borel sets in and is an absolute constant. It has been known that this property of has important consequences for rational approximation: For a compact set , let be the algebra of complex functions on which are uniform limits on of functions analytic in a neighborhood of . Let also be the algebra of those complex functions on which are continuous on and analytic in the interior of . The semi-additivity of implies the old and famous “inner boundary conjecture": If , then . Here is the inner boundary of ; that is, the set of boundary points of that do not belong to the boundary of some component of .
A. M. Davie and B. Øksendal [Acta Math. 149, 127-152 (1982; Zbl 0527.31001)] had proved when the Hausdorff dimension of the inner boundary is finite. Recently, Tolsa proved another important result: the semi-additivity of the analytic capacity [Acta Math. 190, 105–149 (2003; Zbl 1060.30031)]. Some ideas and techniques of that paper are used in the present paper but the details are different in an essential way. For the proof the author uses another quantity, the capacity , where the supremum is taken over all Radon measures on with continuous Cauchy transform satisfying . Another notion used in the proof is the Menger curvature of a Radon measure . This quantity, introduced by M. S. Melnikov [Sb. Nath. 186, 827–846 (1995; Zbl 0840.30008)] has played an important role in the recent progress of related problems. The main step towards the semi-additivity of is the proof that both and are comparable with
where (the linear density of at ).