Geometric & Functional Analysis GAFA

, Volume 8, Issue 2, pp 234–242

Optimal young's inequality and its converse: a simple proof

  • F. Barthe

DOI: 10.1007/s000390050054

Cite this article as:
Barthe, F. GAFA, Geom. funct. anal. (1998) 8: 234. doi:10.1007/s000390050054


We give a new proof of the sharp form of Young's inequality for convolutions, first proved by Beckner [Be] and Brascamp-Lieb [BrLi]. The latter also proved a sharp reverse inequality in the case of exponents less than 1. Our proof is simpler and gives Young's inequality and its converse altogether.

Copyright information

© Birkhäuser Verlag, Basel 1998

Authors and Affiliations

  • F. Barthe
    • 1
  1. 1.Equipe d'Analyse et de Mathématiques Appliquées, Université de Marne-la-Vallée, Cité Descartes, 5 boulevard Descartes, Champs-sur Marne, 77454 Marne-la-Vallée CEDEX 2, France, e-mail: FR

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