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

We consider the spectrum of the Laplacian in a bounded open domain of ℝ
 n
 with a rough boundary (ie with possibly non-integer dimension) and we discuss a conjecture by M V Berry generalizing Weyl's conjecture Then using ideas Mark Kac developed in his famous study of the drum, we give upper and lower bounds for the second term of the expansion of the partition function The main thesis of the paper is to show that the relevant measure of the roughness of the boundary should be based on Minkowski dimensions and on Minkowski measures rather than on Haussdorff ones

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Can One Hear the Dimension of a Fractal

where 'HI is the 1-dimensional Hausdorff measure in Rn, c(x, y, z) is the inverse of the radius of the circumcircle of the triangle (x, y, z), that is, following the terminology of [6], the Menger curvature of the triple (x, y, z). A Borel set E C Rn is said to be "purely unrectifiable" if for any Lipschitz function y: R -t R, 1H l(E n -(R)) = 0 whereas it is said to be rectifiable if there exists a countable family of Lipschitz functions yi : R -E Rn such that Hi1(E\ Ui yi(IR)) = 0. It may be seen from this definition that any 1-set E (that is, E Borel and 0 < 7-(t(E) < oo) can be decomposed into two subsets

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Menger curvature and rectifiability

1. Introduction. Let µ be a continuous (i.e., without atoms) positive Radon measure on the complex plane. The truncated Cauchy integral of a compactly supported function f in L p (µ), 1 ≤ p ≤ +∞, is defined by Ꮿ ε f (z) = |ξ −z|>ε f (ξ) ξ − z dµ(ξ), z ∈ C, ε > 0. In this paper, we consider the problem of describing in geometric terms those measures µ for which |Ꮿ ε f | 2 dµ ≤ C |f | 2 dµ, (1) for all (compactly supported) functions f ∈ L 2 (µ) and some constant C independent of ε > 0. If (1) holds, then we say, following David and Semmes [DS2, pp. 7–8], that the Cauchy integral is bounded on L 2 (µ). A special instance to which classical methods apply occurs when µ satisfies the doubling condition µ(2) ≤ Cµ((), for all discs centered at some point of spt(µ), where 2 is the disc concentric with of double radius. In this case, standard Calderón-Zygmund theory shows that (1) is equivalent to Ꮿ * f 2 dµ ≤ C |f | 2 dµ, (2) where Ꮿ * f (z) = sup ε>0 |Ꮿ ε f (z)|. If, moreover, one can find a dense subset of L 2 (µ) for which Ꮿf (z) = lim ε→0 Ꮿ ε f (z) (3) 269 270 XAVIER TOLSA exists a.e. (µ) (i.e., almost everywhere with respect to µ), then (2) implies the a.e. (µ) existence of (3), for any f ∈ L 2 (µ), and |Ꮿf | 2 dµ ≤ C |f | 2 dµ, for any function f ∈ L 2 (µ) and some constant C. For a general µ, we do not know if the limit in (3) exists for f ∈ L 2 (µ) and almost all (µ) z ∈ C. This is why we emphasize the role of the truncated operators Ꮿ ε. Proving (1) for particular choices of µ has been a relevant theme in classical analysis in the last thirty years. Calderón's paper [Ca] is devoted to the proof of (1) when µ is the arc length on a Lipschitz graph with small Lipschitz constant. The result for a general Lipschitz graph was obtained by Coifman, McIntosh, and Meyer in 1982 in the celebrated paper [CMM]. The rectifiable curves , for which (1) holds for the arc length measure µ on …

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L2-boundedness of the Cauchy integral operator for continuous measures

CONTENTSIntroductionChapter I. The analytic capacity of sets § 1. Definition and some properties of analytic capacity § 2. The connection between the capacity of a set and measures § 3. On removable singularities of analytic functions § 4. The analytic C-capacity of sets § 5. Estimates of the coefficients in the Laurent series § 6. The change in the capacity under a conformal transformation of a set Chapter II. The separation of singularities of functions § 1. The construction of a special system of partitions of unity § 2. Integral representations of continuous functions § 3. Separation of singularities § 4. The approximation of functions in parts § 5. Approximation of functions on sets with empty inner boundary § 6. The additivity of capacity for some special partitions of a setChapter III. Estimation of the Cauchy integral § 1. Statement of the result § 2. Estimate of the Cauchy integral § 3. Estimation of the Cauchy integral along a smooth Lyapunov curve § 4. Proof of the principal theorem § 5. Some consequences § 6. A refinement of the Maximum Principle and the capacity analogue to the theorem on density pointsChapter IV. Classification of functions admitting an approximation by rational fractions § 1. Examples of functions that cannot be approximated by rational fractions § 2. A criterion for the approximability of a function § 3. Properties of the second coefficient in the Laurent series § 4. Proof of the principal lemma § 5. Proof of the theorems of § 2Chapter V. The approximation problem for classes of functions § 1. Removal of the poles of approximating functions from the domain of analyticity of the function being approximated § 2. Necessary conditions for the algebras to coincide § 3. A criterion for the equality of the algebras § 4. Geometrical examples § 5. Some problems in the theory of approximationChapter VI. The approximation of functions on nowhere dense sets § 1. The instability of capacity § 2. A capacity criterion for the approximability of functions on nowhere dense sets § 3. Theorems on the approximation of continuous functions in terms of Banach algebras § 4. A capacity characterization of the Mergelyan function and of peak pointsReferences

https://iopscience.iop.org/article/10.1070/RM1967v022n06ABEH003763/pdf

A G Vitushkin

Papers

The analytic capacity of sets in problems of approximation theory