Traces of Sobolev functions and higher integrability

TL;DR

This paper characterizes how increased interior integrability of Sobolev functions enhances the integrability of their boundary traces, using a novel nonlinear extension operator to establish optimal results.

Contribution

It provides a sharp, new characterization of boundary trace integrability improvements for Sobolev functions, introducing a novel nonlinear extension operator.

Findings

Optimal boundary trace integrability results established.

Introduction of a new nonlinear extension or lifting operator.

Enhanced understanding of Sobolev function boundary behavior.

Abstract

We give a sharp characterization of how additional integrability in the interior improves the integrability of boundary traces of $\mathrm{W}^{1,p}$-Sobolev functions. The optimality of our results relies on a novel nonlinear extension or lifting operator.