Square function characterization of weak Hardy spaces

TL;DR

This paper introduces a new square function characterization for weak Hardy spaces $H^{p,ty}$ across all $p>0$, using interpolation techniques to handle the space's distributional nature.

Contribution

It provides the first square function characterization of $H^{p,ty}$ spaces for all $p>0$, overcoming the challenge of the space's lack of a dense subspace.

Findings

Established a new characterization for weak Hardy spaces.

Extended the understanding of $H^{p,ty}$ spaces beyond previous results.

Utilized interpolation methods to address distributional complexities.

Abstract

We obtain a new square function characterization of the weak Hardy space $H^{p,\infty}$ for all $p\in(0,\iy)$. This space consists of all tempered distributions whose smooth maximal function lies in weak $L^p$. Our proof is based on interpolation between $H^p$ spaces. The main difficulty we overcome is the lack of a good dense subspace of $H^{p,\infty}$ which forces us to work with general $H^{p,\infty}$ distributions.