Sharp weighted log-Sobolev inequalities: characterization of equality cases and applications
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2022
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916132338794496 |
|---|---|
| author | Balogh, Zoltán M. Don, Sebastiano Kristály, Alexandru |
| author_facet | Balogh, Zoltán M. Don, Sebastiano Kristály, Alexandru |
| contents | By using optimal mass transport theory, we provide a direct proof to the sharp $L^p$-log-Sobolev inequality $(p\geq 1)$ involving a log-concave homogeneous weight on an open convex cone $E\subseteq \mathbb R^n$. The perk of this proof is that it allows to characterize the extremal functions realizing the equality cases in the $L^p$-log-Sobolev inequality. The characterization of the equality cases is new for $p\geq n$ even in the unweighted setting and $E=\mathbb R^n$. As an application, we provide a sharp weighted hypercontractivity estimate for the Hopf-Lax semigroup related to the Hamilton-Jacobi equation, characterizing also the equality cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_05578 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Sharp weighted log-Sobolev inequalities: characterization of equality cases and applications Balogh, Zoltán M. Don, Sebastiano Kristály, Alexandru Analysis of PDEs Functional Analysis 28A25, 46E35, 49Q22 By using optimal mass transport theory, we provide a direct proof to the sharp $L^p$-log-Sobolev inequality $(p\geq 1)$ involving a log-concave homogeneous weight on an open convex cone $E\subseteq \mathbb R^n$. The perk of this proof is that it allows to characterize the extremal functions realizing the equality cases in the $L^p$-log-Sobolev inequality. The characterization of the equality cases is new for $p\geq n$ even in the unweighted setting and $E=\mathbb R^n$. As an application, we provide a sharp weighted hypercontractivity estimate for the Hopf-Lax semigroup related to the Hamilton-Jacobi equation, characterizing also the equality cases. |
| title | Sharp weighted log-Sobolev inequalities: characterization of equality cases and applications |
| topic | Analysis of PDEs Functional Analysis 28A25, 46E35, 49Q22 |
| url | https://arxiv.org/abs/2202.05578 |