Kernels of trace operators via fine continuity

Abstract

We study traces of elements of fractional Sobolev spaces $H_p^\alpha(\mathbb{R}^n)$ on closed subsets $\Gamma$ 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 $\Gamma$ if and only if they vanish $\mu$-almost everywhere on $\Gamma$. 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 $\Gamma$ as the closure of $C_c^\infty(\mathbb{R}^n\setminus \Gamma)$ in $H_p^\alpha(\mathbb{R}^n)$. The measures do not have to satisfy a doubling condition. In particular, the set $\Gamma$ may be a finite union of closed sets having different Hausdorff dimensions. We provide corresponding results for fractional Sobolev spaces $H_p^\alpha(\Omega)$ on domains $\Omega\subset \mathbb{R}^n$ satisfying the measure density condition.