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Gaussian kernels have only Gaussian maximizers. (English) Zbl 0726.42005

Author’s abstract: “A Gaussian integral kernel G(x,y) on \({\mathbb{R}}^ n\times {\mathbb{R}}^ n\) 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 \(L^ p({\mathbb{R}}^ n)\) to \(L^ q({\mathbb{R}}^ n)\) and to prove that the \(L^ p({\mathbb{R}}^ n)\) functions that saturate the bound are necessarily Gaussians. This is accomplished generally for \(1<p\leq q<\infty\) and also for \(p>q\) for 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.” This is an excellent paper both in depth of the results obtained and their presentation.

MSC:

42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
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References:

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