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Sibirskie Èlektronnye Matematicheskie Izvestiya [Siberian Electronic Mathematical Reports], 2020, Volume 17, Pages 1552–1570
DOI: https://doi.org/10.33048/semi.2020.17.108
(Mi semr1302)
 

This article is cited in 1 scientific paper (total in 1 paper)

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
References:
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.
Received August 9, 2019, published September 28, 2020
Bibliographic databases:
Document Type: Article
UDC: 517.51
MSC: 46E35, 31C45
Language: English
Citation: I. M. Tarasova, V. A. Shlyk, “Weighted Sobolev spaces, capacities and exceptional sets”, Sib. Èlektron. Mat. Izv., 17 (2020), 1552–1570
Citation in format AMSBIB
\Bibitem{TarShl20}
\by I.~M.~Tarasova, V.~A.~Shlyk
\paper Weighted Sobolev spaces, capacities and exceptional sets
\jour Sib. \`Elektron. Mat. Izv.
\yr 2020
\vol 17
\pages 1552--1570
\mathnet{http://mi.mathnet.ru/semr1302}
\crossref{https://doi.org/10.33048/semi.2020.17.108}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000575249700001}
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  • https://www.mathnet.ru/eng/semr1302
  • https://www.mathnet.ru/eng/semr/v17/p1552
  • This publication is cited in the following articles:
    1. Yu. V. Dymchenko, V. A. Shlyk, “Capacities of generalized condensers with $A_1$-Muckenhoupt weight”, Sib. elektron. matem. izv., 19:1 (2022), 164–186  mathnet  crossref  mathscinet
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
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