New function spaces of BMO type, the John‐Nirenberg inequality, interpolation, and applications

Abstract In this paper, we introduce and develop some new function spaces of BMO (bounded mean oscillation) type on spaces of homogeneous type or measurable subsets of spaces of homogeneous type. The new function spaces are defined by variants of maximal functions associated with generalized approximations to the identity, and they generalize the classical BMO space. We show that the John‐Nirenberg inequality holds on these spaces and they interpolate with L p spaces by the complex interpolation method. We then give applications on L p ‐boundedness of singular integrals whose kernels do not satisfy the Hörmander condition. © 2005 Wiley Periodicals, Inc.