New Orlicz-Hardy Spaces Associated with Divergence Form Elliptic Operators

Abstract

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