  In this paper, we consider the weighted Hardy space $\mathcal{H}^p(\omega)$
induced by an $A_1$ weight $\omega.$ We characterize the positive Borel measure
$\mu$ such that the identical operator maps $\mathcal{H}^p(\omega)$ into
$L^q(d\mu)$ boundedly when $0<p, q<\infty.$ As an application, we obtain
necessary and sufficient conditions for the boundedness of generalized area
operators $A_{\mu,\nu}$ from $\mathcal{H}^p(\omega)$ to $L^q(\omega).$
