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

Let $0<p\le 1$ 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 $R_j,j=1,2,\mathrm{.}\mathrm{.}\mathrm{.},n$, are bounded on $H_w^p({\mathbb{R}}^n)$. In this note we extend this to the general case of weight $w$ in the Muckenhoupt class $A_{\mathrm{\infty }}$ through molecular characterization. One difficulty, which has not been taken care in \cite{LLY}, consists in passing from atoms to all functions in $H_w^p({\mathbb{R}}^n)$. Furthermore, the $H_w^p$-boundedness of $\theta$-Calder'on-Zygmund operators are also given through molecular characterization and atomic decomposition.