Hardy's operator minus identity and power weights

TL;DR

This paper establishes optimal bounds for the Hardy operator minus the identity in weighted function spaces, providing new insights into their norm relations and extending understanding for various classes of functions.

Contribution

It introduces the first sharp bounds for the operator H - I with power weights across different function classes, enhancing the theoretical framework of weighted inequalities.

Findings

Derived optimal bounds for H - I in power-weighted spaces.

Identified the relationships between norms of H and its dual.

Extended results to positive decreasing and general functions.

Abstract

Let $H$ be the Hardy operator and $I$ the identity operator acting on functions on the real half-line. We find optimal bounds for the operator $H - I$ in the setting of power weights and the cases of positive decreasing functions, positive functions, and general functions. As a byproduct, we obtain some results about the optimal relations between the norms of $H$ and its dual.