Hardy Space Estimates for Littlewood-Paley-Stein Square Functions and Calder\'on-Zygmund Operators

TL;DR

This paper establishes new conditions for the boundedness of Littlewood-Paley-Stein square functions and Calderón-Zygmund operators on Hardy spaces with small exponents, expanding the understanding of these operators in low-regularity settings.

Contribution

It introduces novel Carleson measure and polynomial growth BMO conditions that ensure boundedness of these operators on Hardy spaces for all exponents down to zero.

Findings

Carleson measure conditions ensure boundedness of square functions on Hardy spaces.

Polynomial growth BMO conditions guarantee Calderón-Zygmund operators are bounded.

Construction of Bony paraproducts bounded on Hardy spaces with exponents approaching zero.

Abstract

In this work, we give new sufficient conditions for a Littlewood-Paley-Stein square function and necessary and sufficient conditions for a Calder\'on-Zygmund operator to be bounded on Hardy spaces $H^p$ with indices smaller than $1$. New Carleson measure type conditions are defined for Littlewood-Paley-Stein operators, and we show that they are sufficient for the associated square function to be bounded from $H^p$ into $L^p$. New polynomial growth $BMO$ conditions are also introduced for Calder\'on-Zygmund operators. These results are applied to prove that Bony paraproducts can be constructed such that they are bounded on Hardy spaces with exponents ranging all the way down to zero.