Kernels of trace operators via fine continuity

We study traces of elements of fractional Sobolev spaces $H_p^{\alpha }({\mathbb{R}}^n)$ on closed subsets $\mathrm{\Gamma }$ of ${\mathbb{R}}^n$, given as the supports of suitable measures $\mu$. We prove that if these measures satisfy localized upper density conditions, then quasi continuous representatives vanish quasi everywhere on $\mathrm{\Gamma }$ if and only if they vanish $\mu$-almost everywhere on $\mathrm{\Gamma }$. We use this result to characterize the kernel of the trace operator mapping from $H_p^{\alpha }({\mathbb{R}}^n)$ into the space of $\mu$-equivalence classes of functions on $\mathrm{\Gamma }$ as the closure of $C_c^{\mathrm{\infty }}({\mathbb{R}}^n\setminus \mathrm{\Gamma })$ in $H_p^{\alpha }({\mathbb{R}}^n)$. The measures do not have to satisfy a doubling condition. In particular, the set $\mathrm{\Gamma }$ may be a finite union of closed sets having different Hausdorff dimensions. We provide corresponding results for fractional Sobolev spaces $H_p^{\alpha }(\mathrm{\Omega })$ on domains $\mathrm{\Omega }\subset {\mathbb{R}}^n$ satisfying the measure density condition.