The boundedness of multilinear Calder\'on-Zygmund operators on weighted and variable Hardy spaces

TL;DR

This paper proves the boundedness of multilinear Calderón-Zygmund operators on weighted and variable Hardy spaces, extending previous results and introducing a finite atomic decomposition theorem for these spaces.

Contribution

It generalizes boundedness results to weighted and variable Hardy spaces and provides a new atomic decomposition theorem for weighted Hardy spaces.

Findings

Boundedness established for multilinear Calderón-Zygmund operators on weighted Hardy spaces.

Extension of results to variable Hardy spaces.

Introduction of a finite atomic decomposition theorem for weighted Hardy spaces.

Abstract

We establish the boundedness of the multilinear Calder\'on-Zygmund operators from a product of weighted Hardy spaces into a weighted Hardy or Lebesgue space. Our results generalize to the weighted setting results obtained by Grafakos and Kalton (Collect. Math. 2001) and recent work by the third author, Grafakos, Nakamura, and Sawano. As part of our proof we provide a finite atomic decomposition theorem for weighted Hardy spaces, which is interesting in its own right. As a consequence of our weighted results, we prove the corresponding estimates on variable Hardy spaces. Our main tool is a multilinear extrapolation theorem that generalizes a result of the first author and Naibo (Differential Integral Equations 2016).