Complex interpolation and Calderón–Mityagin couples of Morrey spaces

Abstract

We study interpolation spaces between global Morrey spaces and between local Morrey spaces. We prove that for a wide class of couples of these spaces the upper complex Calderón spaces are not described by the $K$-method of interpolation. A by-product of our results is that couples of Morrey spaces belonging to this class are not Calderón–Mityagin couples. A Banach couple $\left(X_0,X_1\right)$ is said to have the universal $K$-property if all relative interpolation spaces from any Banach couple to $\left(X_0,X_1\right)$ are relatively $K$-monotone. A couple of local Morrey spaces is proved to have the universal $K$-property once it is a Calderón–Mityagin couple.