New Orlicz-Hardy Spaces Associated with Divergence Form Elliptic Operators

Abstract

Let $L$ be the divergence form elliptic operator with complex bounded measurable coefficients, $ω$ the positive concave function on $(0,\infty)$ of strictly critical lower type $p_\oz\in (0, 1]$ and $ρ(t)={t^{-1}}/ω^{-1}(t^{-1})$ for $t\in (0,\infty).$ In this paper, the authors study the Orlicz-Hardy space $H_{ω,L}({\mathbb R}^n)$ and its dual space $\mathrm{BMO}_{ρ,L^\ast}({\mathbb R}^n)$, where $L^\ast$ denotes the adjoint operator of $L$ in $L^2({\mathbb R}^n)$. Several characterizations of $H_{ω,L}({\mathbb R}^n)$, including the molecular characterization, the Lusin-area function characterization and the maximal function characterization, are established. The $ρ$-Carleson measure characterization and the John-Nirenberg inequality for the space $\mathrm{BMO}_{ρ,L}({\mathbb R}^n)$ are also given. As applications, the authors show that the Riesz transform $\nabla L^{-1/2}$ and the Littlewood-Paley $g$-function $g_L$ map $H_{ω,L}({\mathbb R}^n)$ continuously into $L(ω)$. The authors further show that the Riesz transform $\nabla L^{-1/2}$ maps $H_{ω,L}({\mathbb R}^n)$ into the classical Orlicz-Hardy space $H_ω({\mathbb R}^n)$ for $p_ω\in (\frac{n}{n+1},1]$ and the corresponding fractional integral $L^{-γ}$ for certain $γ>0$ maps $H_{ω,L}({\mathbb R}^n)$ continuously into $H_{\widetildeω,L}({\mathbb R}^n)$, where $\widetildeω$ is determined by $ω$ and $γ$, and satisfies the same property as $ω$. All these results are new even when $ω(t)=t^p$ for all $t\in (0,\infty)$ and $p\in (0,1)$.