Weak type estimates of Marcinkiewicz integrals on the weighted Hardy and Herz-type Hardy spaces

TL;DR

This paper establishes weighted weak type estimates for Marcinkiewicz integrals on weighted Hardy and Herz-type Hardy spaces using atomic decomposition techniques.

Contribution

It provides new weighted weak type bounds for Marcinkiewicz integrals on specific Hardy spaces, expanding harmonic analysis tools.

Findings

Weighted weak type estimates obtained for Marcinkiewicz integrals.

Application of atomic decomposition theory to these estimates.

Results extend understanding of harmonic analysis on weighted spaces.

Abstract

The Marcinkiewicz integral is essentially a Littlewood-Paley $g$-function, which plays a important role in harmonic analysis. In this article, by using the atomic decomposition theory of weighted Hardy spaces and homogeneous weighted Herz-type Hardy spaces, we will obtain some weighted weak type estimates for Marcinkiewicz integrals on these spaces.