The preduals of Banach space valued Bourgain-Morrey spaces

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.