Generalized Besov-type and Triebel-Lizorkin-type spaces

Let 0 < p < oo, 0 < q <= infinity, and s is an element of R. We introduce a new type of generalized Besov-type spaces B-p,q(s,phi) (R-d) and generalized Triebel-Lizorkin-type spaces F-p,q(s,phi) (R-d), where phi belongs to the class Gp, that is, phi : (0, infinity) -> (0, infinity) is nondecreasing and t(-d/p)phi(t) is nonincreasing in t > 0. We establish several properties of these spaces, including some embedding properties. We also obtain the atomic decomposition of the spaces B-p,q(s,phi) (Rd) and F-p,q(s,phi) (R-d).