Positive Gaussian Kernels also Have Gaussian Minimizers

We study lower bounds on multilinear operators with Gaussian kernels acting on Lebesgue spaces, with exponents below one. We put forward natural conditions when the optimal constant can be computed by inspecting centered Gaussian functions only, and we give necessary and sufficient conditions for th...

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Bibliographic Details
Main Authors: Barthe, Franck, Wolff, Paweł
Format: eBook Book
Language:English
Published: Providence, Rhode Island American Mathematical Society 2022
Edition:1
Series:Memoirs of the American Mathematical Society
Subjects:
Gaussian processes
Inequalities (Mathematics)
Integral operators
Kernel functions
Operator theory > Integral, integro-differential, and pseudodifferential operators > Integral operators. msc
Real functions > Inequalities > Inequalities for sums, series and integrals. msc
optimal transport
Brascamp-Lieb inequality
Gaussian kernel
reversed Gaussian hypercontractivity
positivity improving property
ISBN:9781470451431, 1470451433
ISSN:0065-9266, 1947-6221
Online Access:Get full text
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Summary:We study lower bounds on multilinear operators with Gaussian kernels acting on Lebesgue spaces, with exponents below one. We put forward natural conditions when the optimal constant can be computed by inspecting centered Gaussian functions only, and we give necessary and sufficient conditions for this constant to be positive. Our work provides a counterpart to Lieb’s results on maximizers of multilinear operators with real Gaussian kernels, also known as the multidimensional Brascamp-Lieb inequality. It unifies and extends several inverse inequalities.
Bibliography:Includes bibliographical references (p. 89-90)
March 2022, volume 276, number 1359 (seventh of 7 numbers)
ISBN:9781470451431
1470451433
ISSN:0065-9266
1947-6221
DOI:10.1090/memo/1359