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

It is proved that the existence of the trace operator $Tr\colon B_1^{n-\alpha}\to L^1_E(\mathcal H_\alpha)$, $0\leqslant\alpha<n$, 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 $\mathbb R^n$.