A generalization of John–Nirenberg's inequailty

In work the generalization $BMO_\varphi$ of space of $BMO$ for functions on space of gomogeneous type that defined by integral $\varphi$-oscillations is studied. The analog of John–Nirenberg's inequality for functions from these classes is proved. As a corollary we prove coincidence of the classes $BMO_\varphi$ for rather wide class of functions $\varphi$. Furthermore, generalizations of Kampanato–Meyers's and Spanne's theorems are obtained.