Asymptotic Behaviour of three fractional spaces

We obtain asymptotically sharp identification of fractional Sobolev spaces Wˢp,q, extension spaces Eˢp,q, and Triebel-Lizorkin spaces Ḟˢp,q. In particular we obtain for Wˢp,q and Eˢp,q a stability theory a la Bourgain-Brezis-Mironescu as s → 1, answering a question raised by Brazke--Schikorra--Yung. Part of the results are new even for $p=q$.