This article is cited in 2 papers

Real, complex and functional analysis

Weighted Sobolev spaces, capacities and exceptional sets

I. M. Tarasova, V. A. Shlyk

Vladivostok Branch of Russian Customs Academy, 16v, Strelkovaya str., Vladivostok, 690034, Russia

Abstract: We consider the weighted Sobolev space $W^{m,p}_\omega (\Omega)$, where $\Omega$ is an open subset of $R^n$, $n\ge2$, and $\omega$ is a Muckenhoupt $A_p$-weight on $R^n$, $1\le p<\infty$, $m\in\mathbb N$. For the equalities $W^{m,p}_\omega (\Omega\setminus E)=W^{m,p}_\omega(\Omega)$, $W^{m,p}_\omega(\Omega\setminus E)=W^{m,p}_\omega(\Omega)$ to hold, conditions are obtained in terms of $E$ as a set of zero $(p,m,\omega)$-capacity, or an $NC_{p,\omega}$-set for the first equality. For the equality $W^{m,p}(\Omega)=W^{m,p}(\Omega)$, the conditions are established for $R^n \setminus\Omega$ as a set of zero $(p,m,\omega)$-capacity. Similar results are partially true for $W^m_{p,\omega}(\Omega)$, $L^m_{p,\omega}(\Omega)$.

Keywords: Sobolev space, capacity, Muckenhoupt weight, exceptional set.

UDC: 517.51

MSC: 46E35, 31C45

Received August 9, 2019, published September 28, 2020

Language: English

DOI: 10.33048/semi.2020.17.108

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