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

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