Hypercontractivity of Poisson Semigroups with Orthogonal Polynomial Eigenfunctions

For any $1<p<q<\mathrm{\infty }$, 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 $\alpha ,\beta \ge -1\mathrm{/}2$, the associated Poisson semigroups are ultracontractive, namely bounded from $L^1$ to $L^{\mathrm{\infty }}$. 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.