Inventiones mathematicae

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

Gaussian kernels have only Gaussian maximizers

  • Elliott H. Lieb


A Gaussian integral kernelG(x, y) onR n ×R n 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 fromL p (R n ) toL p (R n ) and to prove that theL p (R n ) 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.


Fourier Fourier Transform Quadratic Form Gaussian Kernel Proof Technique 
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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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