Gaussian kernels have only Gaussian maximizers

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 L^p(Rⁿ) to L^p(Rⁿ) and to prove that the L^p(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.