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Sobolev type embeddings in the limiting case

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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 k/kn as a special case. We deal with generalized Sobolev spaces W kA , 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.

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Authors and Affiliations

  1. Department of Mathematics, Technion-Israel Institute of Technology, 32000, Haifa, Israel

    Michael Cwikel

  2. Department of Mathematics, Technion, Israel Institute of Technology, 32000, Haifa, Israel

    Evgeniy Pustylnik

Authors
  1. Michael Cwikel
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  2. Evgeniy Pustylnik
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Additional information

In memory of Gene Fabes.

Acknowledgements and Notes This research was supported by Technion V.P.R. Fund-M. and C. Papo Research Fund.

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Cwikel, M., Pustylnik, E. Sobolev type embeddings in the limiting case. The Journal of Fourier Analysis and Applications 4, 433–446 (1998). https://doi.org/10.1007/BF02498218

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  • DOI: https://doi.org/10.1007/BF02498218

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