Inventiones mathematicae

, Volume 102, Issue 1, pp 179–208 | Cite as

Gaussian kernels have only Gaussian maximizers

  • Elliott H. Lieb
Article

Abstract

A Gaussian integral kernelG(x, y) onRn×Rn is the exponential of a quadratic form inx andy; the Fourier transform kernel is an example. The problem addressed here is to find the sharp bound ofG as an operator fromLp(Rn) toLp(Rn) and to prove that theLp(Rn) functions that saturate the bound are necessarily Gaussians. This is accomplished generally for 1<pq<∞ and also forp>q in some special cases. Besides greatly extending previous results in this area, the proof technique is also essentially different from earlier ones. A corollary of these results is a fully multidimensional, multilinear generalization of Young's inequality.

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Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • Elliott H. Lieb
    • 1
  1. 1.Department of MathematicsPrinceton UniversityPrincetonUSA

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