Optimal Weighted Poincar\'e and Log-Sobolev Inequalities for Cauchy Measures
Zhengliang Zhang, Bin Qian (IMT), Wei Liu
TL;DR
This paper derives optimal weighted Poincaré and Log-Sobolev inequalities specifically for Cauchy distributions, enhancing understanding of their functional inequalities.
Contribution
It introduces the first optimal weighted inequalities for Cauchy measures, providing precise weight functions for these inequalities.
Findings
Established optimal weighted Poincaré inequalities for Cauchy measures.
Derived sharp Log-Sobolev inequalities with optimal weights.
Enhanced the theoretical understanding of functional inequalities for heavy-tailed distributions.
Abstract
In this paper, We establish the weighted Poincar\'{e} inequalities and Log-Sobolev inequalities for Cauchy distributions with optimal weight functions.
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Taxonomy
TopicsGeometric Analysis and Curvature Flows · Nonlinear Partial Differential Equations · Geometry and complex manifolds