Besov–Bourgain–Morrey–Campanato Spaces: Boundedness of Operators, Duality, and Sharp John–Nirenberg Inequality

Bourgain–Morrey spaces, introduced by J. Bourgain, play an important role in the analysis of some linear and nonlinear partial differential equations. In this article, by exploiting the exquisite geometrical structure of shifted dyadic systems in the Euclidean space, we introduce (dyadic) Besov–Bourgain–Morrey–Campanato spaces via innovatively mixing together both the integral means from Campanato spaces and the structural framework of Besov–Bourgain–Morrey spaces (a recent generalization of Bourgain–Morrey spaces). We then study their fundamental real-variable properties, including the triviality and the nontriviality, their relations with other known function spaces, their predual spaces, as well as sharp John–Nirenberg type inequalities with distinct necessary and sufficient conditions which are different from the case of BMO and Campanato spaces. In particular, after establishing an equivalent quasi-norm of non-dyadic Besov–Bourgain–Morrey–Campanato spaces expressed via integrals, we characterize the boundedness of both Calderón–Zygmund operators and generalized fractional integrals on these non-dyadic function spaces and their predual spaces via vanishing conditions.