Bourgain-Morrey spaces meet structure of Triebel-Lizorkin spaces

Let 0 < q = p = 8, r. (0,8], and Mp q,r (Rn) denote the Bourgain-Morrey space which was introduced by J. Bourgain and has proved important in the study of some linear and nonlinear partial differential equations. In this article, via cleverly combining both the structure of Bourgain-Morrey spaces and the structure of Triebel-Lizorkin spaces and adding an extra exponent t. (0,8], the authors introduce a new class of function spaces, called Triebel-Lizorkin-Bourgain-Morrey spacesM. F p,t q,r (Rn). The authors showthatM. F (p),(t) (q),(r) (R-n) is just a bridge connecting Bourgain-Morrey spaces and global Morrey spaces. In addition, by fully using exquisite geometrical properties of cubes of Euclidean spaces, the authors also explore various fundamental real-variable properties of M. F p, (t) (q),(r) (R-n) as well as its relations with other Morrey type spaces, such as Besov-Bourgain-Morrey spaces and local Morrey spaces. Finally, via finding an equivalent quasi-norm of Herz spaces and making full use of both the Calderon product and the sparse family of dyadic grids, the authors obtain the sharp boundedness on M. F (p), (t) (q),(r) (R-n) of classical operators including the Hardy-Littlewood maximal operator, the Calderon-Zygmund operator, and the fractional integral.