$(P,Q)$ complex hypercontractivity

TL;DR

This paper extends complex hypercontractivity results for polynomials under the Hermite semigroup from real to complex parameters, providing new conditions and applications using heat semigroup techniques.

Contribution

It introduces a comprehensive criterion for $(P,Q)$ complex hypercontractivity with complex parameters, extending Hariya's real parameter results and applying heat semigroup methods.

Findings

Derived necessary and sufficient conditions for complex hypercontractivity.

Extended Hariya's real parameter results to complex parameters.

Presented multiple applications for different polynomial transformations.

Abstract

Let $\xi$ be the standard normal random vector in $\mathbb{R}^{k}$. Under some mild growth and smoothness assumptions on any increasing $P, Q : [0, \infty) \mapsto [0, \infty)$ we show $(P,Q)$ complex hypercontractivity $$ Q^{-1}(\mathbb{E} Q(|T_{z}f(\xi)|))\leq P^{-1}(\mathbb{E}P(|f(\xi)|)) $$ holds for all polynomials $f:\mathbb{R}^{k} \mapsto \mathbb{C}$, where $T_{z}$ is the hermite semigroup at complex parameter $z, |z|\leq 1$, if and only if \begin{align*} \left|\frac{tP''(t)}{P'(t)}-z^{2}\frac{tQ''(t)}{Q'(t)}+z^{2}-1\right|\leq \frac{tP''(t)}{P'(t)}-|z|^{2}\frac{tQ''(t)}{Q'(t)}+1-|z|^{2} \end{align*} holds for all $t>0$ provided that $F''>0$, and $F'/F''$ is concave, where $F = Q\circ P^{-1}$. This extends Hariya's result from real to complex parameter $z$. Several old and new applications are presented for different choices of $P$ and $Q$. The proof uses heat semigroup arguments, where we find a certain map $C(s)$, which interpolates the inequality at the endpoints. The map $C(s)$ itself is composed of four heat flows running together at different times.