# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
The semiadditivity of continuous analytic capacity and the inner boundary conjecture. (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 $E\subset ℂ$ is defined as $\alpha \left(E\right)=sup|{f}^{\text{'}}\left(\infty \right)|$, where the supremum is taken over all functions continuous in $ℂ$, analytic in $ℂ\setminus E$ and satisfying $|f\left(z\right)|\le 1$. For a general $F\subset ℂ$, one defines $\alpha \left(F\right)=sup\left\{\alpha \left(E\right):E\subset F$, $E$ 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:

$\alpha \left(\bigcup _{j=1}^{\infty }{E}_{j}\right)\le c\sum _{j=1}^{\infty }\alpha \left({E}_{j}\right),$

where ${E}_{j}$ are Borel sets in $ℂ$ 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 $ℂ\setminus E$.

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 ${\parallel C\mu \parallel }_{{L}^{\infty }\left(ℂ\right)}\le 1$. 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

$sup\left\{\mu \left(E\right):supp\mu \subset E,\phantom{\rule{0.277778em}{0ex}}{{\Theta }}_{\mu }\left(z\right)=0,\forall z\in E,\phantom{\rule{0.277778em}{0ex}}{c}^{2}\left(\mu \right)\le \mu \left(E\right)\right\},$

where ${{\Theta }}_{\mu }\left(z\right)={lim}_{r\to 0}\mu \left(\left\{\zeta :|\zeta -z| (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