Traces of functions belonging to Sobolev and Besov spaces and extensions from subsets of Euclidean space

A. B. Gulisashvili

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XVI, Tome 157 (1987), pp. 137-145

Résumé

It is proved that the existence of the trace operator $Tr\colon B_1^{n-\alpha}\to L^1_E(\mathcal H_\alpha)$, $0\leqslant\alpha$, implies the existence of the bounded extension (nonlinear) $\mathrm {Ext}\colon L^1(\mathcal H_\alpha)\to B_1^{n-\alpha}$, where $\mathcal H_\alpha$ denotes the $\alpha$-dimensional Hausdorff measure in $\mathbb R^n$ and $E$ is a Borel subset of $\mathbb R^n$.