Lusin Area Function and Molecular Characterizations of Musielak-Orlicz Hardy Spaces and Their Applications

Abstract

Lusin Area Function and Molecular Characterizations of Musielak-Orlicz Hardy Spaces and Their ApplicationsLet $\ddot{I}†:{\mathbb{R}}^n\times [0,\infty )\to [0,\infty )$ be a growth function such that $\ddot{I}†(x,\cdot )$ is nondecreasing, $\ddot{I}†(x,0)=0$, $\ddot{I}†(x,t)>0$ when $t>0$, ${\lim }_{t\to \infty }\ddot{I}†(x,t)=\infty$, and $\ddot{I}†(\cdot ,t)$ is a Muckenhoupt $A_{\infty }({\mathbb{R}}^n)$ weight uniformly in $t$. In this paper, the authors establish the Lusin area function and the molecular characterizations of the Musielak-Orlicz Hardy space $H_{\ddot{I}}†({\mathbb{R}}^n)$ introduced by Luong Dang Ky via the grand maximal function. As an application, the authors obtain the $\ddot{I}†$-Carleson measure characterization of the Musielak-Orlicz $\mathrm{BMO}$-type space ${\mathrm{BMO}}_{\ddot{I}}†({\mathbb{R}}^n)$, which was proved to be the dual space of $H_{\ddot{I}}†({\mathbb{R}}^n)$ by Luong Dang Ky.