Traces of Besov and Lizorkin--Triebel spaces to thick subsets of ${\mathbb{R}}^n$

Let $d\in [0,n]$ and let $E\subset {\mathbb{R}}^n$ be a closed $d$-thick set. Given $p\in [1,\mathrm{\infty }]$, $q\in (0,\mathrm{\infty }]$, and $s\in (\frac{n-d}{p},1)$, we provide an intrinsic description of the trace-space of the Besov space $B_{p,q}^s({\mathbb{R}}^n)$ to $E$. If, in addition, $p<\mathrm{\infty }$, we give an intrinsic description of the trace-space of the Lizorkin–Triebel space $F_{p,q}^s({\mathbb{R}}^n)$ to $E$.