A complete description of traces on $mathbb{R}^{n}$ of functions from the weighted Sobolev space $W^{l}_{1}(mathbb{R}^{n+1},gamma)$, $l in mathbb{N}$, with weight $gamma in A^{rm loc}_{1}(mathbb{R}^{n+1})$ is obtained. In the case $l=1$ the proof of
the trace theorems is based on a~special nonlinear algorithm for constructing a~system of tilings of the space~$mathbb R^n$. As the trace of the space $W^1_1(mathbb R^{n+1},gamma)$ we have the new function space $Z({gamma_{k,m}})$.
The paper is concerned with Besov spaces of variable smoothness $B^{varphi_{0}}_{p,q}(mathbb{R}^{n},{t_{k}})$, in which the norms are defined in terms of convolutions with smooth functions. A relation is found between the spaces $B^{varphi_{0}}_{p,q}
(mathbb{R}^{n},{t_{k}})$ and the spaces $widetilde{B}^{l}_{p,q,r}(mathbb{R}^{n},{t_{k}})$, which were introduced earlier by the author.
The present paper is concerned with new Besov-type space of variable smoothness. Nonlinear spline-approximation approach is used to give atomic decomposition of such space. Characterization of the trace space on hyperplane is also obtained.
