Precompactness in matrix weighted Bourgain-Morrey spaces

In this paper, we introduce matrix weighted Bourgain-Morrey spaces and obtain two sufficient conditions for precompact sets in matrix weighted Bourgain-Morrey spaces. We prove that the dyadic average operator is bounded on some matrix weighted Bourgain-Morrey spaces. With this result, we obtain the necessity for precompact sets in some matrix weighted Bourgain-Morrey spaces. The results are new even for the unweighted Bourgain-Morrey spaces.