A Functorial Approach to Multi-Space Interpolation with Function Parameters

We develop a functorial interpolation theory for infinite families of Banach spaces associated with Boyd-function parameters. This framework unifies two classical extensions of real interpolation - interpolation with function parameters and interpolation of multiple spaces- by providing a common functorial framework encompassing both interpolation with function parameters and interpolation of multiple spaces.
Within this unified setting, we establish the equivalence between the associated K- and J-methods and investigate the resulting interpolation functors. In particular, we prove a power theorem together with several stability and reiteration results. Finally, we illustrate the scope of the theory by identifying interpolation spaces for generalized Sobolev and Lorentz spaces, obtaining generalized Besov spaces as interpolation spaces and deriving reiteration formulas for interpolation of multiple spaces.