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arXiv · 2202.05578

Sharp weighted log-Sobolev inequalities: characterization of equality cases and applications

Zoltán M. Balogh·Sebastiano Don·Alexandru Kristály
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Abstract

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.

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Zoltán M. Balogh, Sebastiano Don, Alexandru Kristály. 2022-02-11. Sharp weighted log-Sobolev inequalities: characterization of equality cases and applications. https://arxiv.org/abs/2202.05578

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