Weak Poincar\'e Inequality for Convolution Probability Measures

TL;DR

This paper establishes weak Poincaré inequalities for certain probability measures on manifolds and convolutions on , providing explicit results and concrete examples to deepen understanding of measure concentration.

Contribution

It introduces Lyapunov-based conditions to derive weak Poincare9 inequalities for measures on manifolds and convolutions on , with explicit examples and applications.

Findings

Weak Poincare9 inequalities are established for measures on manifolds.

Convolution measures on  satisfy weak Poincare9 inequalities under certain conditions.

Explicit examples illustrate the applicability of the theoretical results.

Abstract

By using Lyapunov conditions, weak Poincar\'e inequalities are established for some probability measures on a manifold $(M,g)$. These results are further applied to the convolution of two probability measures on $\R^d$. Along with explicit results we study concrete examples.