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JOURNALS // Sibirskii Matematicheskii Zhurnal // Archive

Sibirsk. Mat. Zh., 2010 Volume 51, Number 6, Pages 1298–1315 (Mi smj2162)

This article is cited in 11 papers

Sufficiency of broken lines in the modulus method and removable sets

Yu. V. Dymchenko, V. A. Shlyk

Far Eastern National University, Vladivostok

Abstract: We establish the sufficiency of the family of broken lines in calculating the modulus of a condenser. We extend the Ahlfors–Beurling definition of removable sets basing on rectangles to weighted Sobolev spaces with a Muckenhoupt weight. We obtain exact characteristics of removable sets in terms of girth by broken lines. We prove the invariance of weighted Sobolev spaces under quasi-isometric mappings.

Keywords: modulus of a family of curves, condenser capacity, Muckenhoupt weight, removable set, Sobolev space, quasi-isometry.

UDC: 517.54+517.554

Received: 29.12.2009
Revised: 17.05.2010


 English version:
Siberian Mathematical Journal, 2010, 51:6, 1028–1042

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