A new approach to norm inequalities on weighted and variable Hardy spaces

TL;DR

This paper presents simplified proofs of Hardy space inequalities for fractional and singular integral operators on weighted and variable Hardy spaces, using atomic decompositions, vector-valued inequalities, and extrapolation.

Contribution

It introduces a new, more straightforward proof technique for Hardy space estimates, enhancing understanding and simplifying existing complex proofs.

Findings

Simplified proofs of Hardy space inequalities

Effective use of atomic decompositions and vector-valued inequalities

Enhanced clarity in the proof of Hardy space estimates

Abstract

We give new proofs of Hardy space estimates for fractional and singular integral operators on weighted and variable exponent Hardy spaces. Our proofs consist of several interlocking ideas: finite atomic decompositions in terms of $L^\infty$ atoms, vector-valued inequalities for maximal and other operators, and Rubio de Francia extrapolation. Many of these estimates are not new, but we give new and substantially simpler proofs, which in turn significantly simplifies the proofs of the Hardy spaces inequalities.