Hypercontractivity of Poisson Semigroups with Orthogonal Polynomial Eigenfunctions
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915899015954432 |
|---|---|
| author | Hormozi, Mahdi Zhu, Jie-Xiang |
| author_facet | Hormozi, Mahdi Zhu, Jie-Xiang |
| contents | 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 $α, β\ge -1/2$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_28223 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Hypercontractivity of Poisson Semigroups with Orthogonal Polynomial Eigenfunctions Hormozi, Mahdi Zhu, Jie-Xiang Classical Analysis and ODEs 47D07, 60J35, 42C05 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 $α, β\ge -1/2$, 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. |
| title | Hypercontractivity of Poisson Semigroups with Orthogonal Polynomial Eigenfunctions |
| topic | Classical Analysis and ODEs 47D07, 60J35, 42C05 |
| url | https://arxiv.org/abs/2603.28223 |