A sharp logarithmic condition for the Hardy operator on $L^{1}(0,\infty)$ and $\ell^1$

TL;DR

This paper introduces a modified Hardy operator with a logarithmic integrability condition that precisely characterizes the largest subspace of functions on which the operator is bounded, addressing a key limitation of the classical Hardy operator.

Contribution

It provides a sharp, logarithmic criterion for the boundedness of a modified Hardy operator on $L^1$ and $\, ext{l}^1$, filling a gap in understanding its behavior on integrable functions.

Findings

Characterization of the largest subspace where the modified Hardy operator is bounded.

Identification of a sharp logarithmic integrability condition.

Extension of Hardy operator analysis to new functional spaces.

Abstract

The Hardy operator is not bounded on the space of integrable functions on the positive half-line and its discrete counterpart on summable sequences. we introduce a modified Hardy operator obtained by subtracting a natural corrective term, and characterize the largest subspace of integrable functions on which this modified operator maps into integrable functions. The sharp condition is a logarithmic integrability (summability) requirement whose weight reflects obstructions on both small and large scales.