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Inequalities pp 417-439 | Cite as

Best Constants in Young’s Inequality, Its Converse, and Its Generalization to More than Three Functions

  • Herm Jan Brascamp
  • Elliott H. Lieb

Abstract

The best possible constant Dmt in the inequality | ∬ dx dyf(x)g(x —y) h(y)|< Dpgt||f||p||g||Q||h||t, p,q,t>|, 1/p + llq+ 1/t = 2, is determined; the equality is reached if /, g, and h are appropriate Gaussians. The same is shown to be true for the converse inequality (0 < p, q < 1, t < 0), in which case the inequality is reversed. Furthermore, an analogous property is proved for an integral of k functions over n variables, each function depending on a linear combination of the n variables; some of the functions may be taken to be fixed Gaussians. Two applications are given, one of which is a pr∞f of Nelson’s hypercontractive inequality.

Keywords

Equality Sign Good Constant Converse Inequality Rearrangement Inequality Schwarz Symmetrization 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Herm Jan Brascamp
    • 1
  • Elliott H. Lieb
    • 2
  1. 1.Department of PhysicsPrinceton UniversityPrinceton
  2. 2.Department of Mathematics and PhysicsPrinceton UniversityPrinceton

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