Bourgain–Morrey spaces and their applications to boundedness of operators

Abstract

A class of function spaces related to Bourgain–Morrey spaces was recently introduced. Here, this class is investigated from the viewpoints of harmonic analysis and functional analysis. Specifically, this paper is oriented to certain inclusion, approximation, and interpolation properties as well as the boundedness of operators such as the Hardy–Littlewood maximal operator, fractional integral operators, fractional maximal operators, and singular integral operators. In particular, the embedding result by Bégout and Vargas is refined. Another important feature of this class of function spaces is that it can describe the convolution property. In addition, we give a description of the dual of Bourgain–Morrey spaces and we combine this with existing results to conclude the reflexivity of these spaces.