Some results concerning localization property of generalized Herz, Herz-type Besov spaces and Herz-type Triebel-Lizorkin spaces

Keywords:
generalized Herz space, Herz-type Besov space, Herz-type Triebel-Lizorkin space, localization propertyAbstract
In this paper, based on generalized Herz-type function spaces ˙Kpq(θ)˙Kpq(θ) were introduced by Y. Komori and K. Matsuoka in 2009, we define Herz-type Besov spaces ˙KpqBsβ(θ)˙KpqBsβ(θ) and Herz-type Triebel-Lizorkin spaces ˙KpqFsβ(θ)˙KpqFsβ(θ), which cover the Besov spaces and the Triebel-Lizorkin spaces in the homogeneous case, where θ={θ(k)}k∈Z is a sequence of non-negative numbers θ(k) such that C−12δ(k−j)≤θ(k)θ(j)≤C2α(k−j),k>j, for some C≥1 (α and δ are numbers in R). Further, under the condition mentioned above on θ, we prove that ˙Kpq(θ) and ˙KpqBsβ(θ) are localizable in the ℓq-norm for p=q, and ˙KpqFsβ(θ) is localizable in the ℓq-norm, i.e. there exists φ∈D(Rn) satisfying ∑k∈Znφ(x−k)=1, for any x∈Rn, such that ‖f|E‖≈(∑k∈Zn‖φ(⋅−k)⋅f|E‖q)1/q. Results presented in this paper improve and generalize some known corresponding results in some function spaces.