A wavelet characterization for weighted Hardy Spaces

Abstract

In this paper, we give a wavelet area integral characterization for weighted Hardy spaces $H^p(\omega ),0<p<\infty$, with $\omega \in A_{\infty }$. Our wavelet characterization establishes the identification between$H^p(\omega )$ and $T_2^p(\omega )$, the weighted discrete tent space, for $0<p<\infty$ and $\omega \in A_{\infty }$. This allows us to use all the results of tent spaces for weighted Hardy spaces. In particular, we obtain the isomorphism between $H^p(\omega )$ and the dual space of $H^{p^{\prime }}(\omega )$ where $1<p<\infty$ and $1/p+1/p^{\prime }=1$, and the wavelet and the Carleson measure characterizations of BMO${}_{\omega }$. Moreover, we obtain interpolation between $A_{\infty }$-weighted Hardy spaces $H^{p_1}(\omega )$ and $H^{p_2}(\omega ),1\le p_1<p_2<\infty$.