Musielak–Orlicz Hardy spaces associated with divergence form elliptic operators without weight assumptions

Abstract

Let $L$ be a divergence form elliptic operator with complex bounded measurable coefficients, let $\omega$ be a positive Musielak–Orlicz function on $(0,\infty )$ of uniformly strictly critical lower-type $p_{\omega }\in (0,1]$, and let $\rho (x,t)=t^{-1}/{\omega }^{-1}(x,t^{-1})$ for $x\in {\mathbb{R}}^n$, $t\in (0,\infty )$. In this paper, we study the Musielak–Orlicz Hardy space $H_{\omega ,L}\left({\mathbb{R}}^n\right)$ and its dual space ${\mathrm{BMO}}_{\rho ,L^{*}}\left({\mathbb{R}}^n\right)$, where $L^{*}$ denotes the adjoint operator of $L$ in $L^2\left({\mathbb{R}}^n\right)$. The $\rho$-Carleson measure characterization and the John–Nirenberg inequality for the space ${\mathrm{BMO}}_{\rho ,L}\left({\mathbb{R}}^n\right)$ are also established. Finally, as applications, we show that the Riesz transform $\nabla L^{-1/2}$ and the Littlewood–Paley $g$-function $g_L$ map $H_{\omega ,L}\left({\mathbb{R}}^n\right)$ continuously into $L\left(\omega \right)$.