Bourgain, Brezis and Mironescu theorem for fractional Sobolev spaces with variable exponents

TL;DR

This paper extends the Bourgain--Brezis--Mironescu theorem to fractional Sobolev spaces with variable exponents, demonstrating a new embedding result for regular functions but also showing the limitations of such embeddings in general.

Contribution

It introduces a Bourgain--Brezis--Mironescu-type theorem for fractional Sobolev spaces with variable exponents and analyzes the conditions under which the embedding theorem holds or fails.

Findings

Established a new embedding theorem for regular functions in variable exponent fractional Sobolev spaces.

Proved that the limiting embedding theorem does not hold universally for these spaces.

Identified the regularity conditions necessary for the theorem to be valid.

Abstract

A Bourgain--Brezis--Mironescu-type theorem for fractional Sobolev spaces with variable exponents is established for sufficiently regular functions. We prove, however, that a limiting embedding theorem for these spaces fails to hold in general.