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

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.