, Volume 102, Issue 1, pp 179-208

Gaussian kernels have only Gaussian maximizers

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Abstract

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.

Oblatum 19-XII-1989
Work partially supported by U.S. National Science Foundation grant PHY-85-15288-A03