On the atomic and molecular decomposition of the weighted Hardy spaces

Abstract

The purpose of this article is to give another molecular decomposition for members of weighted Hardy spaces, different from that given by Lee and Lin [J. Funct. Anal.  188 (2002), no. 2, 442-460], and to review some overlooked details. As an application of this decomposition, we obtain the boundedness on $H^{p}_{w}(\mathbb{R}^{n})$ of every bounded linear operator on some $L^{p_0}(\mathbb{R}^n)$ with $1 < p_0 < +\infty$, for all weights $w \in \mathcal{A}_{\infty}$ and all $0 < p \leq 1$ if $1< \frac{r_w -1}{r_w} p_0$, or all $0 < p < \frac{r_w -1}{r_w} p_0$ if $\frac{r_w -1}{r_w} p_0 \leq 1$, where $r_w$ is the critical index of $w$ for the reverse Hölder condition. In particular, the well-known results about boundedness of singular integrals from $H^{p}_w(\mathbb{R}^{n})$ into $L^{p}_{w}(\mathbb{R}^{n})$ and on $H^{p}_{w}(\mathbb{R}^{n})$ for all $w \in \mathcal{A}_{\infty}$ and all $0 < p \leq 1$ are established. We also obtain the $H^{p}_{w^{p}}(\mathbb{R}^{n})$-$H^{q}_{w^{q}}(\mathbb{R}^{n})$ boundedness of the Riesz potential $I_{\alpha}$ for $0 < p \leq 1$, $\frac{1}{q}=\frac{1}{p} - \frac{\alpha}{n}$, and certain weights $w$.