A mass transportation approach for Sobolev inequalities in variable exponent spaces

TL;DR

This paper introduces a mass transportation method to prove Sobolev-Poincaré inequalities in variable exponent spaces, offering a flexible approach that extends to other inequalities and improves existing results like the Sobolev-trace inequality.

Contribution

It presents a novel mass transportation proof for Sobolev inequalities in variable exponent spaces, enhancing the theoretical toolkit and extending applicability.

Findings

Proved Sobolev-Poincaré inequality using mass transportation.

Extended the method to derive Sobolev-trace inequality.

Improved upon previous results by Fan.

Abstract

In this paper we provide a proof of the Sobolev-Poincar\'e inequality for variable exponent spaces by means of mass transportation methods. The importance of this approach is that the method is exible enough to deal with different inequalities. As an application, we also deduce the Sobolev-trace inequality improving the result obtained by Fan.