A note on $H^p_w$-boundedness of Riesz transforms and $\theta$-Calder\'on-Zygmund operators through molecular characterization

TL;DR

This paper extends the boundedness results of Riesz transforms and $ heta$-Calderón-Zygmund operators on weighted Hardy spaces from $A_1$ weights to the more general $A_$ class using molecular characterization.

Contribution

It generalizes previous boundedness results to $A_$ weights and addresses the challenge of extending from atoms to all functions in $H^p_w$.

Findings

Boundedness of Riesz transforms on $H^p_w$ for $A_$ weights.

Boundedness of $ heta$-Calderón-Zygmund operators via molecular characterization.

Extension from atomic to all functions in $H^p_w$.

Abstract

Let $0 < p \leq 1$ and $w$ in the Muckenhoupt class $A_1$. Recently, by using the weighted atomic decomposition and molecular characterization; Lee, Lin and Yang \cite{LLY} (J. Math. Anal. Appl. 301 (2005), 394--400) established that the Riesz transforms $R_j, j=1, 2,...,n$, are bounded on $H^p_w(\mathbb R^n)$. In this note we extend this to the general case of weight $w$ in the Muckenhoupt class $A_\infty$ through molecular characterization. One difficulty, which has not been taken care in \cite{LLY}, consists in passing from atoms to all functions in $H^p_w(\mathbb R^n)$. Furthermore, the $H^p_w$-boundedness of $\theta$-Calder\'on-Zygmund operators are also given through molecular characterization and atomic decomposition.