  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 $Z(\{\gamma_{k,m}\})$.
