Embedding Theorem For Weighted Hardy Spaces into Lebesgue Spaces

TL;DR

This paper characterizes measures and operators that enable bounded embeddings of weighted Hardy spaces into Lebesgue spaces, extending classical results with new conditions for weighted and generalized area operators.

Contribution

It provides necessary and sufficient conditions for measure and operator boundedness in weighted Hardy spaces, advancing understanding of embeddings into Lebesgue spaces.

Findings

Characterization of measures for bounded embeddings of weighted Hardy spaces

Necessary and sufficient conditions for generalized area operators

Extension of classical embedding theorems to weighted contexts

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 $L^q(\omega).$