The Fatou property of block spaces

TL;DR

This paper proves that block spaces, the predual of Morrey spaces, satisfy the Fatou property and characterizes their predual, filling a longstanding gap in the functional analysis literature.

Contribution

It provides the first proof that block spaces have the Fatou property and characterizes their predual, advancing understanding of their structure.

Findings

Block spaces satisfy the Fatou property.

The predual of block spaces is characterized.

This resolves a long-standing open problem.

Abstract

Around thirty years ago, block spaces, which are the predual of Morrey spaces, had been considered. However, it seems that there is no proof that block spaces satisfy the Fatou property. In this paper the Fatou property for block spaces is verified and the predual of block spaces is characterized.