The sharp maximal function on spaces of generalized homogeneous type

The notion of spaces of a generalized homogeneous type is developed in [2]. In this paper, we introduce the sharp maximal function in this general setting, and establish the equivalence of the L-p norms between the sharp maximal function and the Hardy-Littlewood maximal function, as well the John-Nirenberg type inequalities. As applications, we discuss the concepts of the class of BMO functions and Hardy spaces, and apply our BMO-Hardy space theory to the study of singular integral operators. (C) 1997 Academic Press.