Best bounds for the dual Hardy operator minus identity on decreasing functions

TL;DR

This paper determines the exact norm of the dual Hardy operator minus the identity on decreasing functions, providing optimal bounds and confirming a conjecture, with implications for Hardy-type operator analysis.

Contribution

It presents the exact value of the distance from the identity to the dual Hardy operator on decreasing functions, and establishes optimal bounds and comparisons between related operators.

Findings

Exact value of the distance from I to H* on decreasing functions.

Optimal lower bounds for H*-I on the same cone.

Confirmation of a conjecture regarding Hardy-type operators.

Abstract

The distance from the identity operator $I$ to $H^*$, the dual of the Hardy averaging operator, is studied on the cone of nonnegative, nonincreasing functions in Lebesgue space. The exact value is obtained. Optimal lower bounds are also given for difference, $H^*-I$, of these two operators acting on the same cone. A positive answer is given to a conjecture made in ``The norm of Hardy-type oscillation operators in the discrete and continuous settings'' by A. Ben Said, S. Boza, and J. Soria. Preprint, 2024. In addition, a direct comparison, with optimal constants, is given between the operators $H-I$ and $H^*-I$ acting on the cone.