The Grothendieck property in Marcinkiewicz spaces

TL;DR

This paper characterizes the specific concave functions that make Marcinkiewicz spaces into Grothendieck spaces, enhancing understanding of their structural properties.

Contribution

It provides a complete characterization of when Marcinkiewicz spaces are Grothendieck spaces based on the properties of their defining functions.

Findings

Identifies conditions on concave functions for Marcinkiewicz spaces to be Grothendieck.

Completes the classification of Grothendieck properties in these function spaces.

Advances the theory of Banach spaces related to Marcinkiewicz spaces.

Abstract

The main pupose of this paper is to fully characterize continuous concave functions $\psi $ such that the corresponding Marcinkiewicz Banach function space $M_{\psi }$ is a Grothendieck space.