Preliminaries Subnormal operators: The elementary theory Function theory on the unit circle Hyponormal operators Uniform rational approximation Weak-star rational approximation Some structure theory for subnormal operators Bounded point evaluations.

The theory of subnormal operators

Let $\gamma(E)$ be the analytic capacity of a compact set $E$ and let $\gamma_+(E)$ be the capacity of $E$ originated by Cauchy transforms of positive measures. In this paper we prove that $\gamma(E)\approx\gamma_+(E)$ with estimates independent of $E$. As a corollary, we characterize removable singularities for bounded analytic functions in terms of curvature of measures, and we deduce that $\gamma$ is semiadditive, which solves a long standing question of Vitushkin.

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Painlevé's problem and the semiadditivity of analytic capacity

Painleve's problem and the semiadditivity of analytic capacity

Certain discrete `computable' quantities are introduced, and their interconnections and relations with analytic capacity are found out. The concept of curvature of a measure is introduced, which emerges naturally in the computations of the -norm of the Cauchy transform of this measure. A lower bound on the analytic capacity, which uses the measure curvature and which has, to this extent, a geometric nature, is obtained.

http://iopscience.iop.org/article/10.1070/SM1995v186n06ABEH000045/pdf

Analytic capacity: discrete approach and curvature of measure

For a positive measure AL with compact support in the complex plane C and for 1 < t < co, let pt(i) denote the closure in Lt(/i) of the polynomials in one complex variable. The purpose of this paper is to describe pt(A). The main results are the following. First, there exists a Borel partition {Ail' = of the support of Al such that for i ? 1 the space Pt(t LAAi) contains no nontrivial characteristic functions and

Approximation in the mean by polynomials

https://www.ams.org/tran/1972-171-00/S0002-9947-1972-0350009-9/S0002-9947-1972-0350009-9.pdf

A. M. Davie

Papers

Analytic capacity and approximation problems