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Sobolev type embeddings in the limiting case

  • Michael Cwikel
  • Evgeniy Pustylnik
Article

Abstract

We use interpolation methods to prove a new version of the limiting case of the Sobolev embedding theorem, which includes the result of Hansson and Brezis-Wainger for W n k/k as a special case. We deal with generalized Sobolev spaces W A k , where instead of requiring the functions and their derivatives to be in Ln/k, they are required to be in a rearrangement invariant space A which belongs to a certain class of spaces “close” to Ln/k.

We also show that the embeddings given by our theorem are optimal, i.e., the target spaces into which the above Sobolev spaces are shown to embed cannot be replaced by smaller rearrangement invariant spaces. This slightly sharpens and generalizes an, earlier optimality result obtained by Hansson with respect to the Riesz potential operator.

Math Subject Classifications

Primary 46E35 46M35 

Keywords and Phrases

Sobolev embedding theorem Real interpolation method Lorentz-Zygmund space Riesz potential Bessel potential 

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Copyright information

© Birkhäuser Boston 1998

Authors and Affiliations

  • Michael Cwikel
    • 1
  • Evgeniy Pustylnik
    • 2
  1. 1.Department of MathematicsTechnion-Israel Institute of TechnologyHaifaIsrael
  2. 2.Department of Mathematics, TechnionIsrael Institute of TechnologyHaifaIsrael

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