Traces of weighted Sobolev spaces with Muckenhoupt weight. The case $p=1$

A complete description of traces on ${\mathbb{R}}^n$ of functions from the weighted Sobolev space $W_1^l({\mathbb{R}}^{n+1},\gamma )$, $l \in $\mathbb{N}$$,withweight$\gamma \in A^{\rm loc}_{1}($\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 $Z(\{{\gamma }_{k,m}\})$.