Generalized Besov-type and Triebel-Lizorkin-type spaces

Abstract

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