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 $H_{\tilde{\omega },L}({\mathbb{R}}^n)$, where $\tilde{\omega }$ 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 $p\in (0,1)$.

J. Funct. Anal. (to appear)

References in corpus (1)

• Correction to ``Hardy and BMO spaces associated to divergence form elliptic operators"