Weighted Morrey spaces-complex interpolation and the boundedness of the Hardy-Littlewood maximal operator (Harmonic Analysis and Nonlinear Partial Differential Equations)

Abstract

The aim of this paper is to address two difficult problems of the Morrey spaces. One is the complex interpolation and another is the behavior of the Hardy-Littlewood maximal operator. Weighted Morrey spaces are difficult to handle due to the following reasons: 1. They are not reflexive. 2. Unlike Lebesgue spaces, there are many non-trivial closed linear subspaces. 3. The norm of the indicator function of the cubes is difficult to calculate. Nevertheless, it is possible to calculate the second complex interpolation in some special cases. This will allow us to calculate the complex interpolations in such cases. Although we can not always calculate the norm of the indicator function of the cubes, the boundedness of the Hardy-Littlewood maximal operator makes this possible. In this connection, the first half of this artile is devoted to the complex interpolation. In the latter half we investigate what happens if the Hardy-Littlewood maximal operator is bounded on weighted Morrey spaces. As an application, we prove what can we say for the class of weights by using the complex interpolation.