Analytic capacity, rectifiability, and the Cauchy integral

Abstract

Abstract. A compact set E ⊂ C is said to be removable for bounded analytic functions if for any open set Ω containing E, every bounded function analytic on Ω\E has an analytic extension to Ω. Analytic capacity is a notion that, in a sense, measures the size of a set as a non removable singularity. In particular, a compact set is removable if and only if its analytic capacity vanishes. The so called Painlevé problem consists in characterizing removable sets in geometric terms. Recently many results in connection with this very old and challenging problem have been obtained. Moreover, it has also been proved that analytic capacity is semiadditive. We review these results and other related questions dealing with rectifiability, the Cauchy transform, and the Riesz transforms. Mathematics Subject Classification (2000). Primary 30C85; Secondary 42B20, 28A75.

Minimal Hölder regularity implying finiteness of integral Menger curvature

“…exactly the same way as in the one dimensional case (cf. (9) with z 1 0 , z 1 1 and z 1 0 + ζ 1 (z 1 1 − z 1 0 ) playing the roles of x, y and z respectively and with z−x y−x = ζ 1 ) and then obtain (18) F (z 1 0 + ζ 1 (z 1 1 − z 1 0 )) − F (z 1 0 ) − ζ 1 (F (z 1 1 ) − F (z 1 0 )) ≥ C|z 1 1 − z 1 0 | 1+α ≥CN −n(1+α) .…”

“…This was a crucial step in the proof of Vitushkin's conjecture and allowed to fully characterize removable sets of bounded analytical functions. Surveys of Mattila [8] and Tolsa [18] explain in more detail the connection between these subjects.…”

Removable sets for homogeneous linear partial differential equations in Carnot groups

“…The eventual solution to the Painlevé problem was obtained by Tolsa [40] following extensive work by many people including Ahlfors, Denjoy, Vitushkin, Garnett, Calderón, Mattila, Jones, David, Mel'nikov and Verdera. Tolsa's article [41], written for the proceedings of the 2006 ICM, is highly recommended for an informative history and survey on the Painlevé problem.…”

How averaged Menger curvatures control regularity and topology of curves and surfaces

“…the quotient of length and thickness, see Cantarella et al (2002Cantarella et al ( , 2006Cantarella et al ( , 2011Cantarella et al ( , 2012; Gonzalez et al (2002); Gonzalez & de la Llave (2003); Schuricht & von der Mosel (2004), as well as for elastic rods with self-contact Schuricht & von der Mosel (2003), or packing problems as in Gerlach & von der Mosel (2011b,a). One possible generalization of thickness to surfaces including an existence theory for area-minimizing thick surfaces with prescribed genus or in given isotopy classes was published by Strzelecki & von der Mosel (2005, 2006.…”