Asymptotic Behaviour of fractional seminorms

We obtain asymptotically sharp identification of fractional Sobolev spaces $W_{p,q}^s$, extension spaces $E_{p,q}^s$, and Triebel-Lizorkin spaces ${\dot{F}}_{p,q}^s$. In particular we obtain for $W_{p,q}^s$ and $E_{p,q}^s$ a stability theory a la Bourgain-Brezis-Mironescu as $s\to 1$, answering a question raised by Brazke–Schikorra–Yung. Part of the results are new even for $p=q$.