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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Hlavní autori: Barthe, Franck, Wolff, Paweł
Médium: E-kniha Kniha
Jazyk:English
Vydavateľské údaje: Providence, Rhode Island American Mathematical Society 2022
Vydanie:1
Edícia:Memoirs of the American Mathematical Society
Predmet:
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
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Obsah:
  • Introduction -- Well-posedness of the Minimization Problem and the Minimum Value -- Proof of the Main Theorem -- Geometric Brascamp-Lieb Inequality -- Dual Form of Inverse Brascamp-Lieb Inequalities -- Interpolation -- Positivity in the Rank One Case -- Positivity Condition in the General Case
  • Cover -- Title page -- Chapter 1. Introduction -- 1.1. Background and motivation -- 1.2. Notation and main results -- 1.3. Acknowledgments -- Chapter 2. Well-posedness of the Minimization Problem and the Minimum Value -- 2.1. A non-degeneracy condition -- 2.2. Calculations for centered Gaussian functions -- 2.3. Ensuring finiteness for some functions -- 2.4. On the effect of translating Gaussian functions and consequences of positivity -- 2.5. Case analysis and non-degeneracy hypotheses -- Chapter 3. Proof of the Main Theorem -- 3.1. Decomposition of the kernel -- 3.2. More on quadratic forms -- 3.3. Preliminaries and general strategy of the proof -- 3.4. Optimal transport map -- 3.5. Classes of test functions -- 3.6. Transportation argument -- 3.7. Surjectivity of the change of variable map -- 3.8. Approximation argument -- Chapter 4. Geometric Brascamp-Lieb Inequality -- 4.1. Finding the infimum on centered Gaussian functions -- 4.2. Geometric version of Inverse Brascamp-Lieb inequalities -- 4.3. Relation with the results of Chen, Dafnis and Paouris -- Chapter 5. Dual Form of Inverse Brascamp-Lieb Inequalities -- Chapter 6. Interpolation -- Chapter 7. Positivity in the Rank One Case -- 7.1. No kernel -- 7.2. With a kernel -- Chapter 8. Positivity Condition in the General Case -- 8.1. Recursive structure of the problem -- 8.2. Formulation of the characterization result -- 8.3. Useful notation for the proof -- 8.4. Necessity of Condition (C) -- 8.5. Sufficiency of Condition (C) -- Bibliography -- Back Cover