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 theLpnorms 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