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<p≦q<∞ 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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Oblatum 19-XII-1989
Work partially supported by U.S. National Science Foundation grant PHY-85-15288-A03
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Lieb, E.H. Gaussian kernels have only Gaussian maximizers. Invent Math 102, 179–208 (1990). https://doi.org/10.1007/BF01233426
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DOI: https://doi.org/10.1007/BF01233426