Description of interpolation spaces for the pair of local Morrey-type spaces and their generalizations

Abstract

We consider the real interpolation method and prove that for general local Morrey-type spaces, in the case when they have the same integrability parameter, the interpolation spaces are again general local Morrey-type spaces with appropriately chosen parameters. This result is a particular case of the interpolation theorem for much more general spaces defined with the help of an operator acting from some function space to the cone of non-negative non-decreasing functions on $ (0,\infty)$ . It is also shown how the classical interpolation theorems due to Stein-Weiss, Peetre, Calder\'\''{o}n, Gilbert, Lizorkin, Freitag and some of their new variants can be derived from this theorem.