Geodesic
Weighted Sobolev spaces, capacities and exceptional sets
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 1552-1570.

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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. 
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I. M. Tarasova; V. A. Shlyk. Weighted Sobolev spaces, capacities and exceptional sets. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 1552-1570. http://geodesic.mathdoc.fr/item/SEMR_2020_17_a140/

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