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A necessary condition for Sobolev extension domains in higher dimensions (2207.00541v1)
Published 1 Jul 2022 in math.AP, math.CA, and math.FA
Abstract: We give a necessary condition for a domain to have a bounded extension operator from $L{1,p}(\Omega)$ to $L{1,p}(\mathbb Rn)$ for the range $1 < p < 2$. The condition is given in terms of a power of the distance to the boundary of $\Omega$ integrated along the measure theoretic boundary of a set of locally finite perimeter and its extension. This generalizes a characterizing curve condition for planar simply connected domains, and a condition for $W{1,1}$-extensions. We use the necessary condition to give a quantitative version of the curve condition. We also construct an example of an extension domain that is homeomorphic to a ball and has $n$-dimensional boundary.