Embedding Theorem For Weighted Hardy Spaces into Lebesgue Spaces

Abstract

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 $L^q(\omega ).$

20 pages