A note on properties of the restriction operator on Sobolev spaces

TL;DR

This paper investigates the properties of the restriction operator on Sobolev spaces over non-Lipschitz domains, focusing on conditions for injectivity, surjectivity, and isometry, and providing explicit formulas for minimal norm extensions.

Contribution

It offers new insights into the behavior of the restriction operator on Sobolev spaces on non-Lipschitz sets, including explicit formulas for minimal norm extensions.

Findings

Conditions for restriction operator to be injective, surjective, or isometric.

Explicit formula for minimal norm extension in a special case.

Analysis of the relation between global and local distribution spaces.

Abstract

In our companion paper (S.N. Chandler Wilde, D.P. Hewett, A. Moiola, Sobolev spaces on non-Lipschitz subsets of $\mathbb{R}^n$ with application to boundary integral equations on fractal screens, 2016) we studied a number of different Sobolev spaces on a general (non-Lipschitz) open subset $\Omega$ of $\mathbb{R}^n$, defined as closed subspaces of the classical Bessel potential spaces $H^s(\mathbb{R}^n)$ for $s\in\mathbb{R}$. These spaces are mapped by the restriction operator to certain spaces of distributions on $\Omega$. In this note we make some observations about the relation between these spaces of global and local distributions. In particular, we study conditions under which the restriction operator is or is not injective, surjective and isometric between given pairs of spaces. We also provide an explicit formula for minimal norm extension (an inverse of the restriction operator in appropriate spaces) in a special case.