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The functional volume product under heat flow

Shohei Nakamura, Hiroshi Tsuji

Abstract

We prove that the functional volume product for even functions is monotone increasing along the Fokker--Planck heat flow. This in particular yields a new proof of the functional Blaschke--Santaló inequality by K. Ball and also Artstein-Avidan--Klartag--Milman in the even case. This result is the consequence of a new understanding of the regularizing property of the Ornstein--Uhlenbeck semigroup. That is, we establish an improvement of Borell's reverse hypercontractivity inequality for even functions and identify the sharp range of the admissible exponents. As another consequence of successfully identifying the sharp range for the inequality, we derive the sharp $L^p$-$L^q$ inequality for the Laplace transform for even functions. The best constant of the inequality is attained by centered Gaussians, and thus this provides an analogous result to Beckner's sharp Hausdorff--Young inequality. Our technical novelty in the proof is the use of the Brascamp--Lieb inequality for log-concave measures and Cramér--Rao's inequality in this context.

The functional volume product under heat flow

Abstract

We prove that the functional volume product for even functions is monotone increasing along the Fokker--Planck heat flow. This in particular yields a new proof of the functional Blaschke--Santaló inequality by K. Ball and also Artstein-Avidan--Klartag--Milman in the even case. This result is the consequence of a new understanding of the regularizing property of the Ornstein--Uhlenbeck semigroup. That is, we establish an improvement of Borell's reverse hypercontractivity inequality for even functions and identify the sharp range of the admissible exponents. As another consequence of successfully identifying the sharp range for the inequality, we derive the sharp - inequality for the Laplace transform for even functions. The best constant of the inequality is attained by centered Gaussians, and thus this provides an analogous result to Beckner's sharp Hausdorff--Young inequality. Our technical novelty in the proof is the use of the Brascamp--Lieb inequality for log-concave measures and Cramér--Rao's inequality in this context.
Paper Structure (7 sections, 9 theorems, 72 equations)

This paper contains 7 sections, 9 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Heat flow monotonicity of the functional volume product
  3. An improvement of Borell's reverse hypercontractivity
  4. Proof of Theorems \ref{['t:MonoVP']} and \ref{['t:SymmRevHC']}
  5. Concluding remarks
  6. $L^r$-volume product by Berndtsson--Mastrantonis--Rubinstein
  7. Brascamp--Lieb theory view point and Kolesnikov--Werner's conjecture

Key Result

Theorem 1.1

For all even functions $f\colon \mathbb{R}^n \to {\mathbb R}_+$ with $0<\int_{\mathbb{R}^n}f\, dx <+\infty$, The case of equality in e:FBS appears if and only if $f = c \gamma_A$ for some positive definite matrix $A$ and $c>0$.

Theorems & Definitions (14)

  • Theorem 1.1: Ball BallPhd, Artstein-Avidan--Klartag--Milman AKM
  • Theorem 1.2
  • Lemma 1.3
  • Remark
  • Theorem 1.4: Borell Borell
  • Theorem 1.5
  • Corollary 1.6
  • Theorem 2.1
  • proof : Proof of Theorem \ref{['t:MonoVP']}
  • proof : Proof of Theorem \ref{['t:SymmRevHC']}
  • ...and 4 more