Weighted norm inequalities of various square functions and Volterra integral operators on the unit ball

TL;DR

This paper establishes weighted inequalities for various square functions and Volterra integral operators on the complex unit ball, providing new characterizations of weighted Hardy spaces and answering an open question in the field.

Contribution

It proves weighted inequalities for the Lusin area integral and Volterra operators on the unit ball, and characterizes weighted Hardy spaces via the Lusin area integral.

Findings

Weighted inequalities for Lusin area integral with $A_p$ weights

Characterization of weighted Hardy spaces using the Lusin area integral

Weighted inequalities for Volterra integral operators

Abstract

In this paper, we investigate various square functions on the complex unit ball. We prove the weighted inequalities of the Lusin area integral associated with Poisson integral in terms of $A_p$ weights for all $1<p<\infty$; this gives an affirmative answer to an open question raised by Segovia and Wheeden. In addition, we get an equivalent characterization of weighted Hardy spaces by means of the Lusin area integral in the context of holomorphic functions. We also obtain the weighted inequalities for Volterra integral operators.