*(English)*Zbl 1060.30032

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 $E\subset \u2102$ is defined as $\alpha \left(E\right)=sup|{f}^{\text{'}}\left(\infty \right)|$, where the supremum is taken over all functions continuous in $\u2102$, analytic in $\u2102\setminus E$ and satisfying $\left|f\right(z\left)\right|\le 1$. For a general $F\subset \u2102$, one defines $\alpha \left(F\right)=sup\left\{\alpha \right(E):E\subset F$, $E$ 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 ${E}_{j}$ are Borel sets in $\u2102$ and $c$ is an absolute constant. It has been known that this property of $\alpha $ has important consequences for rational approximation: For a compact set $E$, let $R\left(E\right)$ be the algebra of complex functions on $E$ which are uniform limits on $E$ of functions analytic in a neighborhood of $E$. Let also $A\left(E\right)$ be the algebra of those complex functions on $E$ which are continuous on $E$ and analytic in the interior of $E$. The semi-additivity of $\alpha $ implies the old and famous “inner boundary conjecture”: If $\alpha \left({\partial}_{i}\left(E\right)\right)=0$, then $R\left(E\right)=A\left(E\right)$. Here ${\partial}_{i}E$ is the inner boundary of $E$; that is, the set of boundary points of $E$ that do not belong to the boundary of some component of $\u2102\setminus E$.

*A. M. Davie* and *B. Øksendal* [Acta Math. 149, 127-152 (1982; Zbl 0527.31001)] had proved $R\left(E\right)=A\left(E\right)$ when the Hausdorff dimension of the inner boundary is finite. Recently, *Tolsa* proved another important result: the semi-additivity of the analytic capacity $\gamma $ [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 ${\alpha}_{+}\left(E\right):=sup\mu \left(E\right)$, where the supremum is taken over all Radon measures $\mu $ on $E$ with continuous Cauchy transform $C\mu $ satisfying ${\parallel C\mu \parallel}_{{L}^{\infty}\left(\u2102\right)}\le 1$. Another notion used in the proof is the Menger curvature ${c}^{2}\left(\mu \right)$ of a Radon measure $\mu $. 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 $\alpha $ is the proof that both $\alpha \left(E\right)$ and ${\alpha}_{+}\left(E\right)$ are comparable with

where ${{\Theta}}_{\mu}\left(z\right)={lim}_{r\to 0}\mu \left(\right\{\zeta :|\zeta -z|<r\})/r$ (the linear density of $\mu $ at $z$).

##### MSC:

30C85 | Capacity and harmonic measure in the complex plane |

30E10 | Approximation in the complex domain |

30D50 | Blaschke products, etc. (MSC2000) |

42B20 | Singular and oscillatory integrals, several variables |