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

Under certain restrictions on $s,p,q$, the Triebel-Lizorkin spaces can be viewed as generalised fractional Sobolev spaces $W_q^{s,p}$. In this article, we show that the Bourgain-Brezis-Mironescu formula holds for $W_q^{s,p}$-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].