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Spectral Theory, Function Spaces and Inequalities pp 55–67Cite as

A New, Rearrangement-free Proof of the Sharp Hardy–Littlewood–Sobolev Inequality

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Part of the book series: Operator Theory: Advances and Applications ((OT,volume 219))

Abstract

We show that the sharp constant in the Hardy–Littlewood–Sobolev inequality can be derived using the method that we employed earlier for a similar inequality on the Heisenberg group. The merit of this proof is that it does not rely on rearrangement inequalities; it is the first one to do so for the whole parameter range.

Mathematics Subject Classification (2010). Primary 39B62; Secondary 26A33, 26D10, 46E35.

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Authors and Affiliations

  1. Department of Mathematics, Princeton University, Princeton, NJ, 08544, USA

    Rupert L. Frank & Elliott H. Lieb

Authors
  1. Rupert L. Frank
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  2. Elliott H. Lieb
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Corresponding author

Correspondence to Rupert L. Frank .

Editor information

Editors and Affiliations

  1. School of Computer Science, Cardiff University, The Parade 5, Cardiff, CF24 3AA, United Kingdom

    B. Malcolm Brown

  2. Dept. Mathematics, Ohio State University, West 18th Ave. 231, Columbus, 43210-1174, Ohio, USA

    Jan Lang

  3. School of Mathematics, Statistics &, Actuarial Science, University of Kent, Canterbury, Kent, CT2 7NF, United Kingdom

    Ian G. Wood

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Frank, R.L., Lieb, E.H. (2012). A New, Rearrangement-free Proof of the Sharp Hardy–Littlewood–Sobolev Inequality. In: Brown, B., Lang, J., Wood, I. (eds) Spectral Theory, Function Spaces and Inequalities. Operator Theory: Advances and Applications(), vol 219. Springer, Basel. https://doi.org/10.1007/978-3-0348-0263-5_4

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