A Gaussian integral kernel G(x, y) on Rn×Rn is the exponential of a quadratic form in x and y; the Fourier transform kernel is an example. The problem addressed here is to find the sharp bound of G as an operator from Lp(Rn) to Lp(Rn) and to prove that the Lp(Rn) functions that saturate the bound are necessarily Gaussians. This is accomplished generally for 1<p≦q<∞ and also for p>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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