Progress on Hardy-type Inequalities

TL;DR

This paper reviews recent advances in Hardy-type inequalities within Harmonic Analysis, highlighting probabilistic motivations, non-linear cases, and potential for significant improvements in estimates.

Contribution

It provides a comprehensive survey of recent progress, including new analytic proofs and improved estimates for Hardy-type inequalities.

Findings

Probabilistic motivation links Hardy inequalities to $L^2$-theory.

Analytic proof techniques for non-linear Hardy inequalities.

Potential for substantial improvements in basic estimates.

Abstract

This paper surveys some of our recent progress on Hardy-type inequa\-lities which consist of a well-known topic in Harmonic Analysis. In the first section, we recall the original probabilistic motivation dealing with the stability speed in terms of the $L^2$-theory. A crucial application of a result by Fukushima and Uemura (2003) is included. In the second section, the non-linear case (a general Hardy-type inequality) is handled with a direct and analytic proof. In the last section, it is illustrated that the basic estimates presented in the first two sections can still be improved considerably.