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^{[11]} established that the Riesz transforms $R_j$, $j = 1,2, \cdots ,n$, are bounded on $H^p_w(\mathbf{R}^n)$. 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 [11], consists in passing from atoms to all functions in $H^p_w(\mathbf{R}^n)$. Furthermore, the $H^p_w$-boundedness of $\theta$-Calderón-Zygmund operators are also given through molecular characterization and atomic decomposition.