Positive Gaussian Kernels Also Have Gaussian Minimizers.

Format:

Book

Author/Creator:

Barthe, Franck.

Contributor:

Wolff, Paweł.

Series:

Memoirs of the American Mathematical Society

Memoirs of the American Mathematical Society ; v.276

Language:

English

Subjects (All):

Gaussian processes.

Kernel functions.

Inequalities (Mathematics).

Integral operators.

Physical Description:

1 online resource (102 pages)

Edition:

1st ed.

Place of Publication:

Providence : American Mathematical Society, 2022.

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"-- Provided by publisher.

Contents:

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.

Notes:

Description based on publisher supplied metadata and other sources.

Other Format:

Print version: Barthe, Franck Positive Gaussian Kernels Also Have Gaussian Minimizers

ISBN:

9781470470258

147047025X

OCLC:

1312158592