Complex interpolation and Calderón–Mityagin couples of Morrey spaces

Mastyło, Mieczysław; Sawano, Yoshihiro

doi:10.2140/apde.2019.12.1711

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Complex interpolation and Calderón–Mityagin couples of Morrey spaces

Mieczysław Mastyło and Yoshihiro Sawano

Vol. 12 (2019), No. 7, 1711–1740

DOI: 10.2140/apde.2019.12.1711

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 $(X_{0}, X_{1})$ is said to have the universal $K$ -property if all relative interpolation spaces from any Banach couple to $(X_{0}, X_{1})$ 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.

Keywords

interpolation spaces, complex method, $K$-method, Morrey spaces

Mathematical Subject Classification 2010

Primary: 46B70

Secondary: 42A45, 42B30

Milestones

Received: 12 October 2017

Revised: 3 September 2018

Accepted: 19 December 2018

Published: 22 July 2019

Authors

Mieczysław Mastyło
Faculty of Mathematics and Computer Science Adam Mickiewicz University Poznań Poland

Yoshihiro Sawano
Department of Mathematics and Information Sciences Tokyo Metropolitan University Tokyo Japan

Vol. 12, No. 7, 2019