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

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 (Calc Var Partial Differ Equ 62(2):41–33, (2023).