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 beviewed 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 questionraised by Brazke-Schikorra-Yung in [Bourgain-Brezis-Mironescu convergence viaTriebel-Lizorkin spaces; Calc. Var. Partial Differential Equations; 2023].