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This article is cited in 2 scientific papers (total 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.
Received August 9, 2019, published September 28, 2020
Citation:
I. M. Tarasova, V. A. Shlyk, “Weighted Sobolev spaces, capacities and exceptional sets”, Sib. Èlektron. Mat. Izv., 17 (2020), 1552–1570
Linking options:
https://www.mathnet.ru/eng/semr1302 https://www.mathnet.ru/eng/semr/v17/p1552
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