Interpolation of generalized Morrey spaces

Abstract

In this paper, we shall establish a theory of interpolation of generalized Morrey spaces. We use the complex interpolation methods. Our results extend the interpolation results for Morrey spaces which are discussed by Lu et al. (Can Math Bull 57:598–608, 2014), and also Lemarie-Rieusset (2014). We establish the interpolation of generalized weak Morrey spaces, generalized Orlicz–Morrey spaces and generalized weak Orlicz–Morrey spaces. We also consider the closure of the functions which are essentially bounded and have compact support. The second interpolation of such spaces will yield a class of closed spaces; we describe the second interpolation of the closure of the functions which are essentially bounded and have compact support. This result will carry over to generalized Morrey spaces, generalized weak Morrey spaces, generalized Orlicz–Morrey spaces and generalized weak Orlicz–Morrey spaces. We also give several examples that explain the subtlety of proving the interpolation of Morrey spaces.