Traces of differentiable functions on subsets of Euclidean space

We investigate conditions of existence and coincideness of the traces of functions of Sobolev's and Besov's classes both in the operator sense and in the sense of strict definiteness. A solution is given to the problems on trace and extension for the trace operator $\operatorname{Tr}\colon B\to L^p$ in the case, when $\Gamma$ is a countably $(\mathcal H_m,m)$ – rectifiable $\mathcal H$-measurable subset of $\mathbb R^n$.