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

IF 1.2 2区数学 Q1 MATHEMATICS

Transactions of the American Mathematical Society Pub Date : 2024-03-29 DOI:10.1090/tran/9163

Zoltán Balogh, Sebastiano Don, Alexandru Kristály

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引用次数: 0

Abstract

By using optimal mass transport theory, we provide a direct proof to the sharp L p L^p -log-Sobolev inequality ( p ≥ 1 ) (p\geq 1) involving a log-concave homogeneous weight on an open convex cone E ⊆ R n 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 L^p -log-Sobolev inequality. The characterization of the equality cases is new for p ≥ n p\geq n even in the unweighted setting and E = R n 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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尖锐加权对数-索博列夫不等式：平等情况的特征及应用

通过使用最优质量输运理论，我们直接证明了涉及开放凸锥 E ⊆ R n E\subseteq \mathbb R^n 上对数凹同质权的尖锐 L p L^p -log-Sobolev 不等式 ( p ≥ 1 ) (p\geq 1) 。这个证明的好处在于它可以描述实现 L p L^p -log-Sobolev 不等式中相等情形的极值函数。对于 p ≥ n p\geq n，即使是在无权设置和 E = R n E=\mathbb R^n 的情况下，相等情况的特征描述也是新的。作为应用，我们为与汉密尔顿-雅可比方程相关的霍普夫-拉克斯半群提供了一个尖锐的加权超收缩性估计，同时也描述了相等情况。

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来源期刊

Transactions of the American Mathematical Society 数学-数学

CiteScore

2.30

自引率

7.70%

发文量

171

审稿时长

3-6 weeks

期刊介绍： All articles submitted to this journal are peer-reviewed. The AMS has a single blind peer-review process in which the reviewers know who the authors of the manuscript are, but the authors do not have access to the information on who the peer reviewers are. This journal is devoted to research articles in all areas of pure and applied mathematics. To be published in the Transactions, a paper must be correct, new, and significant. Further, it must be well written and of interest to a substantial number of mathematicians. Piecemeal results, such as an inconclusive step toward an unproved major theorem or a minor variation on a known result, are in general not acceptable for publication. Papers of less than 15 printed pages that meet the above criteria should be submitted to the Proceedings of the American Mathematical Society. Published pages are the same size as those generated in the style files provided for AMS-LaTeX.