Hardy and Carleson Measure Spaces Associated with Operators on Spaces of Homogeneous Type

Let (X, d, μ) be a metric measure space with doubling property. The Hardy spaces associated with operators L were introduced and studied by many authors. All these spaces, however, were first defined by L2(X) functions and finally the Hardy spaces were formally defined by the closure of these subspaces of L2(X) with respect to Hardy spaces norms. A natural and interesting question in this context is to characterize the closure. The purpose of this paper is to answer this question. More precisely, we will introduce CMOLp(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${CMO}_{L}^{p}(X)$\end{document}, the Carleson measure spaces associated with operators L, and characterize the Hardy spaces associated with operators L via (CMOLp(X))′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$({CMO}_{L}^{p}(X))'$\end{document}, the distributions of CMOLp(X)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${CMO}_{L}^{p}(X)$\end{document}.