Hypercontractivity of Poisson Semigroups with Orthogonal Polynomial Eigenfunctions

TL;DR

This paper studies the hypercontractive properties of Poisson semigroups linked to specific operators, revealing differences in boundedness and ultracontractivity across various cases and examining stability under subordination.

Contribution

It provides new results on the fixed-time hypercontractivity and ultracontractivity of Poisson semigroups for Ornstein--Uhlenbeck, Laguerre, and Jacobi operators, including stability analysis under subordination.

Findings

Poisson semigroups for Ornstein--Uhlenbeck and Laguerre fail to be bounded from L^p to L^q for fixed t > 0.

Poisson semigroups for Jacobi operators with α, β ≥ -1/2 are ultracontractive, bounded from L^1 to L^∞.

Hypercontractivity is not stable under Bernstein subordination for these semigroups.

Abstract

For any $1 < p < q < \infty$, we investigate fixed-time hypercontractive bounds from $L^p$ to $L^q$ of Poisson semigroups associated with the Ornstein--Uhlenbeck, Laguerre and Jacobi operators. We prove that, in the Ornstein--Uhlenbeck and Laguerre cases, the Poisson semigroups fail to be $L^p \to L^q$ bounded for any fixed $t > 0$. In contrast, for Jacobi operators with $\alpha, \beta \ge -1/2$, the associated Poisson semigroups are ultracontractive, namely bounded from $L^1$ to $L^\infty$. More generally, we study Bernstein subordinations of these semigroups and show that fixed-time hypercontractivity is not stable under subordination. The analysis relies on quantitative $L^q$-estimates for the corresponding orthogonal polynomial eigenfunctions, together with a bilinear test with the exponential family.