New Besov-type space of variable smoothness and the problem of traces for the weighted Sobolev space

For the weighted Sobolev space $W^{l}_{p}(\mathbb{R}^{n},\gamma)$ a complete description of the trace spaces for planes of dimension $1 \le d < n$ is obtained. The weight $\gamma$ depends on all variables and locally satisfies the Muckenhoupt condition. It appears that in the case $1\le r < p <\infty$ the trace space for $W^{l}_{p}(\mathbb{R}^{n},\gamma)$, $\gamma \in A^{loc}_{\frac{p}{r}}(\mathbb{R}^{n})$ is the Besov type space $\widetilde{B}^{l}_{p,p,r}(\mathbb{R}^{d},\{\gamma_{k}\})$ with variable smoothness $\{\gamma_{k}\}$. The norm in $\widetilde{B}^{l}_{p,q,r}(\mathbb{R}^{d},\{\gamma_{k}\})$ is defined with the help of local best approximations in the $L_{r}$-metric.
Various properties of the space $\widetilde{B}^{l}_{p,q,r}(\mathbb{R}^{d},\{\gamma_{k}\})$ are studied by using the method of nonlinear spline approximation for all values of the parameters $0<p,q,r<\infty$, $l \in \mathbb{N}$ under the minimal assumptions on the variable smoothness $\{\gamma_{k}\}$. For example we present the atomic decomposition theorem, embedding theorems and description of the trace space of $\widetilde{B}^{l}_{p,q,r}(\mathbb{R}^{d},\{\gamma_{k}\})$. The space $\widetilde{B}^{l}_{p,q,r}(\mathbb{R}^{d},\{\gamma_{k}\})$ is compared with 2-microlocal Besov space $B^{\{\gamma_{k}\}}_{p,q}(\mathbb{R}^{d})$ intensively studied by many mathematicians.