04322nam 22006133 450 991095812670332120231110212634.09781470470258147047025X(MiAaPQ)EBC6939724(Au-PeEL)EBL6939724(CKB)21420567900041(OCoLC)1312158592(EXLCZ)992142056790004120220327d2022 uy 0engurcnu||||||||txtrdacontentcrdamediacrrdacarrierPositive Gaussian Kernels Also Have Gaussian Minimizers1st ed.Providence :American Mathematical Society,2022.©2022.1 online resource (102 pages)Memoirs of the American Mathematical Society ;v.276Print version: Barthe, Franck Positive Gaussian Kernels Also Have Gaussian Minimizers Providence : American Mathematical Society,c2022 9781470451431 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."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.Memoirs of the American Mathematical Society Gaussian processesKernel functionsInequalities (Mathematics)Integral operatorsReal functions -- Inequalities -- Inequalities for sums, series and integralsmscOperator theory -- Integral, integro-differential, and pseudodifferential operators -- Integral operatorsmscGaussian processes.Kernel functions.Inequalities (Mathematics)Integral operators.Real functions -- Inequalities -- Inequalities for sums, series and integrals.Operator theory -- Integral, integro-differential, and pseudodifferential operators -- Integral operators.519.2/3519.2326D1547G10mscBarthe Franck1799928Wolff Paweł1799929MiAaPQMiAaPQMiAaPQBOOK9910958126703321Positive Gaussian Kernels Also Have Gaussian Minimizers4344357UNINA