Kernels of trace operators via fine continuity

TL;DR

This paper characterizes the kernel of trace operators for fractional Sobolev spaces on irregular sets using measure density conditions, extending previous results to non-doubling measures and sets with varying Hausdorff dimensions.

Contribution

It provides a new characterization of the kernel of trace operators for fractional Sobolev spaces on complex sets with minimal measure assumptions.

Findings

Kernel of trace operator characterized as closure of functions vanishing near the set

Results apply to sets with different Hausdorff dimensions and non-doubling measures

Provides measure density conditions for fractional Sobolev spaces on domains

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.