The functional volume product under heat flow

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.