Besov spaces on closed subsets of $\textbf {R}^ n$

Abstract:

Motivated by the need in boundary value problems for partial differential equations, classical trace theorems characterize the trace to a subset $F$ of ${\mathbb {R}^n}$ of Sobolev spaces and Besov spaces consisting of functions defined on ${\mathbb {R}^n}$, if $F$ is a linear subvariety ${\mathbb {R}^d}$ of ${\mathbb {R}^n}$ or a $d$-dimensional smooth submanifold of ${\mathbb {R}^n}$. This was generalized in [2] to the case when $F$ is a $d$-dimensional fractal set of a certain type. In this paper, traces are described when $F$ is an arbitrary closed set. The result may also be looked upon as a Whitney extension theorem in ${L^p}$.