Description of traces of functions in the Sobolev space with a Muckenhoupt weight

We characterize the trace of the Sobolev space $W_p^l(\mathbb R^n,\gamma)$ with $1<p<\infty$ and weight $\gamma\in A_p^\mathrm{loc}(\mathbb R^n)$ on a $d$-dimensional plane for $1\le d<n$. It turns out that for a function $\varphi$ to be the trace of a function $f\in W_p^l(\mathbb R^n,\gamma)$, it is necessary and sufficient that $\varphi$ belongs to a new Besov space of variable smoothness, $\overline B{}_p^l(\mathbb R^d,\{\gamma_{k,m}\})$, constructed in this paper. The space $\overline B{}_p^l(\mathbb R^d,\{\gamma_{k,m}\})$ is compared with some earlier known Besov spaces of variable smoothness.