On the Bourgain, Brezis, and Mironescu Theorem Concerning Limiting Embeddings of Fractional Sobolev Spaces

The article is concerned with the Bourgain, Brezis and Mironescu theorem on the asymptotic behaviour of the norm of the Sobolev-type embedding operator: Ws,p→Lpn/(n−sp) as s↑1 and s↑n/p. Their result is extended to all values of s∈(0,1) and is supplied with an elementary proof. The relationlims↓0s∫Rn∫Rn∣u(x)−u(y)∣p∣x−y∣n+spdxdy=2p−1∣Sn−1∣∥u∥pLp(Rn) is proved.