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

Zoltán M. Balogh, Sebastiano Don, Alexandru Kristály

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.

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

Abstract

By using optimal mass transport theory, we provide a direct proof to the sharp -log-Sobolev inequality involving a log-concave homogeneous weight on an open convex cone . The perk of this proof is that it allows to characterize the extremal functions realizing the equality cases in the -log-Sobolev inequality. The characterization of the equality cases is new for even in the unweighted setting and . 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.
Paper Structure (15 sections, 8 theorems, 202 equations)

This paper contains 15 sections, 8 theorems, 202 equations.

Table of Contents

  1. Introduction
  2. Preparatory results
  3. Proof of Theorem \ref{['log-Sobolev']}: the case $p>1$
  4. Proof of the inequality \ref{['sharp-log-Sobolev']}: case $p>1$, $\tau>0$
  5. Equality in \ref{['sharp-log-Sobolev']}: case $p>1$, $\tau>0$
  6. Proof of the inequality \ref{['sharp-log-Sobolev']}: case $p>1$, $\tau=0$
  7. Equality in \ref{['sharp-log-Sobolev']}: case $p>1$, $\tau=0$
  8. Proof of Theorem \ref{['log-Sobolev-1']}: the case $p=1$
  9. Proof of the inequality \ref{['sharp-log-Sobolev-1']}: case $p=1$, $\tau>0$
  10. Equality in \ref{['sharp-log-Sobolev-1']}: case $p=1$, $\tau>0$
  11. Proof of the inequality \ref{['sharp-log-Sobolev-1']}: case $p=1$, $\tau=0$
  12. Equality in \ref{['sharp-log-Sobolev-1']}: case $p=1$, $\tau=0$
  13. Application: sharp weighted hypercontractivity
  14. Proof of the inequality \ref{['hyperc-estimate']}
  15. Equality in \ref{['hyperc-estimate']}

Key Result

Theorem 1.1

$($Case $p>1$$)$ Let $E\subseteq \mathbb R^n$ be an open convex cone and $\omega\colon E\to (0,\infty)$ be a log-concave homogeneous weight of class $\mathcal{C}^1$ with degree $\tau\geq 0,$ and $p> 1.$ Then for every function $u\in {W}^{1,p}(\omega;E)$ with $\int_E |u|^p\omega dx=1$ we have where Equality holds in sharp-log-Sobolev if and only if the extremal function belongs to the family of G

Theorems & Definitions (17)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1
  • proof
  • Remark 2.1
  • Proposition 2.2
  • Proposition 3.1
  • proof
  • Remark 3.1
  • ...and 7 more