Bilateral Hardy-type inequalities

TL;DR

This paper advances the understanding of Hardy-type inequalities with two weights on intervals, introducing new bounds and extending results to related inequalities like Nash and Sobolev, with elementary proofs.

Contribution

It introduces new isoperimetric constants for Hardy inequalities and extends the analysis to Nash, Sobolev, and logarithmic Sobolev inequalities with simpler proofs.

Findings

Smaller bounds for Hardy inequalities using new isoperimetric constants

Extension of results to Nash, Sobolev, and logarithmic Sobolev inequalities

Introduction of inequalities motivated by probability theory

Abstract

This paper studies the Hardy-type inequalities on the intervals (may be infinite) with two weights, either vanishing at two endpoints of the interval or having mean zero. For the first type of inequalities, in terms of new isoperimetric constants, the factor of upper and lower bounds becomes smaller than the known ones. The second type of the inequalities is motivated from probability theory and is new in the analytic context. The proofs are now rather elementary. Similar improvements are made for Nash inequality, Sobolev-type inequality, and the logarithmic Sobolev inequality on the intervals.