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 $φ: \mathbb R^n\times [0,\infty)\to[0,\infty)$ be a growth function such that $φ(x,\cdot)$ is nondecreasing, $φ(x,0)=0$, $φ(x,t)>0$ when $t>0$, $\lim_{t\to\infty}φ(x,t)=\infty$, and $φ(\cdot,t)$ 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 $H_φ(\mathbb{R}^n)$ introduced by Luong Dang Ky via the grand maximal function. As an application, the authors obtain the $φ$-Carleson measure characterization of the Musielak-Orlicz ${\mathop\mathrm{BMO}}$-type space $\mathop\mathrm{BMO}_φ(\mathbb{R}^n)$, which was proved to be the dual space of $H_φ(\mathbb{R}^n)$ by Luong Dang Ky.