Duality relations in the theory of analytic capacity

Abstract:

This is a survey of duality relations arising in the theory of analytic capacity and its modifications, namely, the Cauchy capacities with various types of measures. Principal attention is paid to the material not published earlier. Also, new modifications of the capacities mentioned above are treated. Linear extremal problems (such as the problem of calculating the analytic capacity of a set) are dual to approximation processes with size constraints (the size of the approximants and the rate of approximation are measured in terms of different metrics). For the new versions of capacity introduced in this paper, approximation with size constraints is extended to the case where the approximants are taken from a fixed conical wedge (rather than from a linear subspace, as it has always been before). Extremely peculiar approximation processes arise via duality from the modifications of the analytic capacity the definition of which involves positive measures (a typical example is the so-called “positive” analytic capacity $\gamma ^{+}$). More precisely, in such situations it is not even required to find an approximant within a small distance from a given element. Instead, in a fixed subspace or conical wedge we seek an addition that brings the above element to a fixed cone. Moreover, this addition must be as small as possible in a certain sense.

Extension of the collection of relations for the analytic capacity and its modifications, and comparison of various relations of this sort make it possible to better understand exceptional sets arising in various branches of holomorphic function theory. In particular, some information is obtained about possible approximation processes on null-sets in the sense of a particular capacity.