We show that the class of weights $w$ for which the Calderon operator is bounded on $L^p(w)$ can be used to develop a theory of real interpolation which is more general and exhibits new features when compared to the usual variants of the Lions-Peetre methods. In particular we obtain extrapolation theorems (in the sense of Rubio de Francia's theory) and reiteration theorems for these methods. We also consider interpolation methods associated with the classes of weights for which the Calderon operator is bounded on weighted Lorentz spaces and obtain similar results. We extend the commutator theorems associated with the real method of interpolation in several directions. We obtain weighted norm inequalities for higher order commutators as well as commutators of fractional order.One application of our results gives new weighted norm inequalities for higher order commutators of singular integrals with multiplications by BMO functions. We also introduce analogs of the space BMO in order to consider the relationship between commutators for Calderon type operators and their corresponding classes of weights.
Introduction Calderon weights Applications to real interpolation: reiteration and extrapolation Other classes of weights Extrapolation of weighted norm inequalities via extrapolation theory Applications to function spaces Commutators defined by the K-method Generalized commutators The quasi Banach case Applications to harmonic analysis BMO type spaces associated to Calderon weights Atomic decompositions and duality References.