  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.
