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

  • Maz'ya, V.
  • Shaposhnikova, T.
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Publication date
November 2002
Publisher
Elsevier Science (USA).

Abstract

AbstractThe 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

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