In this article, we establish a new atomic decomposition for
$f\,\in \,L_{w}^{2}\,\bigcap \,H_{w}^{p}$, where the decomposition converges in
$L_{w}^{2}$-norm rather than in the distribution sense. As applications of this decomposition, assuming that
$T$ is a linear operator bounded on
$L_{w}^{2}$ and
$0\,<\,p\,\le \,1$, we obtain (i) if
$T$ is uniformly bounded in
$L_{w}^{p}$-norm for all
$w-p$-atoms, then
$T$ can be extended to be bounded from
$H_{w}^{p}$ to
$L_{w}^{2}$; (ii) if
$T$ is uniformly bounded in
$H_{w}^{p}$-norm for all
$w-p$-atoms, then
$T$ can be extended to be bounded on
$H_{w}^{p}$; (iii) if
$T$ is bounded on
$H_{w}^{p}$, then
$T$ can be extended to be bounded from
$H_{w}^{p}$ to
$L_{w}^{2}$.