Precompactness in matrix weighted Bourgain-Morrey spaces

TL;DR

This paper introduces matrix weighted Bourgain-Morrey spaces, establishes conditions for precompactness, and proves the boundedness of the dyadic average operator, providing new insights even for unweighted spaces.

Contribution

It presents the first sufficient and necessary conditions for precompactness in matrix weighted Bourgain-Morrey spaces and analyzes the dyadic average operator's boundedness.

Findings

Established sufficient conditions for precompact sets.

Proved boundedness of the dyadic average operator.

Results are new even for unweighted Bourgain-Morrey spaces.

Abstract

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.