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

Under certain restrictions on s, p, q, the Triebel-Lizorkin spaces can be viewed as generalised fractional Sobolev spaces \(W^{s,p}_q\) $W_q^{s,p}$ . In this article, we show that the Bourgain-Brezis-Mironescu formula holds for \(W^{s,p}_q\) $W_q^{s,p}$ -seminorms in arbitrary domain. This addresses an open question raised by Brazke-Schikorra-Yung (Calc Var Partial Differ Equ 62(2):41–33, (2023).