Asymptotic Behaviour of three fractional spaces

We obtain asymptotically sharp identification of fractional Sobolev spaces W^s₏, ₐ, extension spaces E^s₏, ₐ, and Triebel-Lizorkin spaces Ḟˢ₏, ₐ. In particular we obtain for W^s₏, ₐ and E^s₏, ₐ 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.