This article is cited in 1 paper

Real, complex and functional analysis

Removable sets for Sobolev spaces with Muckenhoupt $A_1$-weight

V. A. Shlyk<sup>ab

a</sup> Vladivostok Branch of Russian Customs Academy, 16v, Strelkovaya str., Vladivostok, 690034, Russia
^b Institute of Applied Mathematics, Vladivostok Branch of the RAS, 7, Radio str., Vladivostok, 690041, Russia

Abstract: Let $\Omega$ be an open set in $R^n$, $n\ge2$, and $E$ be a relatively closed subset of $\Omega$. In this paper we obtain a criterion of equality $L^1_{1,\omega}(\Omega\setminus E)=L^1_{1,\omega}(\Omega)$ in terms of $E$ as an $NC_{1,\omega}$-set in $\Omega$ with $A_1$-weight $\omega$. In addition, we establish exact characterizations of $NC_{1,\omega}$-sets in terms of $NED_{1,\omega}$-sets and of the $(1,\omega)$-girth condition. In the case $\omega\equiv1$, these results complete the studies of Vodop'yanov and Gol'dstein on removable sets for $L^1_p(\Omega)$, $p\in(1,+\infty)$.

Keywords: Sobolev space, capacity and modulus of condenser, Muckenhoupt weight, removable set.

UDC: 517.51

MSC: 46E35, 31C45

Received September 9, 2020, published March 3, 2021

Language: English

DOI: 10.33048/semi.2021.18.012

Bibliographic databases:

•