John–Nirenberg Type Inequalities for Musielak–Orlicz Campanato Spaces on Spaces of Homogeneous Type

Abstract

Let X be a space of homogeneous type in the sense of Coifman and Weiss. Let φ : X × [0, ∞) → [0, ∞) be such that φ(x,⋅) is an Orlicz function and φ(⋅, t) is a Muckenhoupt A∞(X) weight uniformly in t. In this paper, we propose John–Nirenberg type inequalities for Musielak–Orlicz Campanato spaces on spaces of homogeneous type. As an application, we show the coincidence between the space BMO(X) and the weighted space BMOw(X) whenever w ∈ A∞(X).