Let $\Omega$ be a John domain, and let $\Gamma\subset \partial \Omega$ be an $h$-set. For some functions $h$ and some weight functions depending on distance from $\Gamma$, embedding theorems for a weighted Sobolev class is obtained.
Widths of weighted Sobolev classes on a John domain
Estimates for the Kolmogorov widths of weighted Sobolev classes with conditions on the $0$th and the greatest derivatives
Quasihyperbolic geodesics in John domains in R^n
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