Weak-type operators and the strong fundamental lemma of real interpolation theory

We prove an interpolation theorem for weak-type operators. This is closely related to interpolation between weak-type classes. Weak-type classes at the ends of interpolation scales play a similar role to that played by BMO with respect to the L-p interpolation scale. We also clarify the roles of some of the parameters appearing in the definition of the weak-type classes. The interpolation theorem follows from a K-functional inequality for the operators, involving the Calderon operator. The inequality was inspired by a K-J inequality approach developed by Jawerth and Milman. We show that the use of the Calderon operator is necessary. We use a new version of the strong fundamental lemma of interpolation theory that does not require the interpolation couple to be mutually closed.