VMO spaces associated with divergence form elliptic operators

Consider the second order divergence form elliptic operator L with complex bounded coefficients. In this paper we introduce and develop a function space VMOL of vanishing mean oscillation associated with the operator L. Using the theory of tent spaces and the Littlewood-Paley theory, we prove that the Hardy space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H_{L}^1}$$\end{document} of Hofmann and Mayboroda introduced in Hofmann and Mayboroda (Math Ann 344:37–116, 2009) is the dual of our \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\rm VMO}_{L^{\ast}}}$$\end{document}, in which L* is the adjoint operator of L. We also give an equivalent characterization of the space VMOL in the context of the theory of tent spaces.