Local Besov–Bourgain–Morrey Spaces: Nontriviality, Boundedness of Local Operators, Duality, and Interpolation

Let 0 < q <= p <= r <= infinity and tau is an element of (0, infinity]. In this article, we introduce a local variant MBq,r p,tau of Besov-Bourgain-Morrey spaces M B-center dot (p,tau) (q,r) , whose special case tau = r was originally introduced by J. Bourgain and has proved to play an important role in the study related to the Strichartz estimate and some partial differential equations. These local spaces MBp,tau (q,r) include Bourgain-Lebesgue, local Morrey, and amalgam spaces as special cases. We find the sufficient and necessary conditions, respectively, for their nontriviality, for MB center dot p,tau.(q,r) to be properly contained in MB (p,tau) (q,r) , and also for both the boundedness and the Fefferman-Stein vector-valued maximal inequality about the local Hardy-Littlewood maximal operator on MB (p,tau)(q,r). Moreover, on MBp,tau (q,r) we study their diversity, their duality, and their interpolation in terms of Calder & oacute;n products. Using the Calder & oacute;n product and a new pointwise sparse domination, we obtain the boundedness of local fractional integrals from Calder & oacute;n products to MBp,tau (q,r) . Moreover, we also obtain the boundedness of local Calder & oacute;n-Zygmund operators on MB (p,tau) (q,r ).