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

Extending results in \cite{M} and \cite{MM} we characterize the classical classes of weights that satisfy reverse H\"{o}lder inequalities in terms of indices of suitable families of $K-$functionals of the weights. In particular, we introduce a Samko type of index (cf. \cite{kara}) for families of functions, that is based on quasi-monotonicity, and use it to provide an index characterization of the $RH_{p}$ classes, as well as the limiting class $RH=$ $RH_{LLogL}=$. $\bigcup\limits_{p>1}RH_{p}$ (cf. \cite{BMR}),\ which in the abstract case involves extrapolation spaces. Reverse H\"{o}lder inequalities associated to $L(p,q)$ norms, and non-doubling measures are also treated.