Reverse Hölder inequalities revisited : Interpolation, Extrapolation, Indices and Doubling

Abstract

Extending results in [50] and [49] we characterize the classical classes of weights that satisfy reverse Hölder inequalities in terms of indices of suitable families of K?functionals of the weights. In particular, we introduce a Samko type of index (cf. [41]) for families of functions, that is based on quasi-monotonicity, and use it to provide an index characterization of the RHp classes, as well as the limiting class RH = RHLLogL =. S p>1 RHp (cf. [8]), which in the abstract case involves extrapolation spaces. Reverse Hölder inequalities associated to L(p, q) norms, and non-doubling measures are also treated.