Bourgain-Brezis-Mironescu formula for $W^{s,p}_q$-spaces in arbitrary domains

TL;DR

This paper proves that the Bourgain-Brezis-Mironescu formula applies to generalized fractional Sobolev spaces $W^{s,p}_q$ in any domain, extending previous results and answering an open question in the field.

Contribution

It establishes the validity of the Bourgain-Brezis-Mironescu formula for $W^{s,p}_q$-spaces in arbitrary domains, broadening the understanding of fractional Sobolev spaces.

Findings

Bourgain-Brezis-Mironescu formula holds for $W^{s,p}_q$-seminorms in any domain

Addresses an open question from recent literature

Extends the applicability of fractional Sobolev space theory

Abstract

Under certain restrictions on $s,p,q$, the Triebel-Lizorkin spaces can be viewed as generalised fractional Sobolev spaces $W^{s,p}_q$. In this article, we show that the Bourgain-Brezis-Mironescu formula holds for $W^{s,p}_q$-seminorms in arbitrary domain. This addresses an open question raised by Brazke-Schikorra-Yung in [Bourgain-Brezis-Mironescu convergence via Triebel-Lizorkin spaces; Calc. Var. Partial Differential Equations; 2023].