We show that the class of weights w for which the Calderon operator is bounded on
Lp(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 ex-
tend the commutator theorems associated with the real method of interpolation in
several directions. We obtain weighted norm inequalities for higher order commuta-
tors 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.
2000 Mathematics Subject Classification. Primary 46B70, 46E30. Secondary 42B25.
Key words and phrases. Calderon operator, weighted norm inequalities, interpolation of op-
erators, commutators, BMO.