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

Samson Owusu-Ensaw, Benoit F. Sehba, Ransford T. Tweneboanah

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.

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

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.
Paper Structure (5 sections, 8 theorems, 50 equations)

This paper contains 5 sections, 8 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. The result for the continuous Hardy operator
  3. The case of the discrete Hardy operator
  4. Conclusion
  5. Compliance with Ethical Standards

Key Result

Theorem 1.1

HL1930 Let $a>0$. Assume that $f$ is a positive measurable function defined on $(0,\infty)$. Then the following hold.

Theorems & Definitions (14)

  • Theorem 1.1
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Theorem 2.3
  • proof : Proof of Theorem \ref{['thm:main1']}
  • Corollary 2.4
  • Definition 3.1
  • Theorem 3.2: Hardy, 1920
  • Proposition 3.3
  • ...and 4 more