The preduals of Banach space valued Bourgain-Morrey spaces

TL;DR

This paper introduces Banach space valued Bourgain-Morrey spaces, characterizes their preduals as block spaces, and explores their duality, reflexivity, and boundedness properties of maximal operators.

Contribution

It establishes the predual relationship between block spaces and Bourgain-Morrey spaces in the Banach space setting, extending classical results.

Findings

Block spaces are the predual of Bourgain-Morrey spaces.

The paper proves completeness, denseness, and Fatou property of block spaces.

It demonstrates the reflexivity of these vector-valued spaces.

Abstract

Let $X$ be a Banach space such that there exists a Banach space $^\ast X$ and $ ( ^\ast X )^ \ast = X $. In this paper, we introduce $X$-valued Bourgain-Morrey spaces. We show that $^\ast X$-valued block spaces are the predual of $X$-valued Bourgain-Morrey spaces. We obtain the completeness, denseness and Fatou property of $^\ast X$-valued block spaces. We give a description of the dual of $X$-valued Bourgain-Morrey spaces and conclude the reflexivity of these spaces. The boundedness of powered Hardy-Littlewood maximal operator in vector valued block spaces is obtained.