The functional volume product under heat flow

TL;DR

This paper proves the monotonicity of the functional volume product under heat flow for even functions, providing new proofs of classical inequalities and establishing sharp bounds for related integral transforms.

Contribution

It introduces a novel approach using the Ornstein--Uhlenbeck semigroup and sharp inequalities, extending classical results to the even function setting.

Findings

Monotonicity of the functional volume product under heat flow.

New proof of the functional Blaschke--Santaló inequality for even functions.

Sharp $L^p$-$L^q$ inequality for the Laplace transform with Gaussian extremizers.

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--Santal\'{o} 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 Cram\'{e}r--Rao's inequality in this context.