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

Authors

  • A. Djeriou Laboratory of Mathematics Pure and Applied, M'Sila University, P.O. Box 166, M'Sila 28000, Algeria
  • R. Heraiz Laboratory of Mathematics Pure and Applied, M'Sila University, P.O. Box 166, M'Sila 28000, Algeria
https://doi.org/10.15330/cmp.13.1.217-228

Keywords:

generalized Herz space, Herz-type Besov space, Herz-type Triebel-Lizorkin space, localization property
Published online: 2021-06-30

Abstract

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)}kZ is a sequence of non-negative numbers θ(k) such that C12δ(kj)θ(k)θ(j)C2α(kj),k>j, for some C1 (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 q-norm for p=q, and ˙KpqFsβ(θ) is localizable in the q-norm, i.e. there exists φD(Rn) satisfying kZnφ(xk)=1, for any xRn, such that f|E(kZnφ(k)f|Eq)1/q. Results presented in this paper improve and generalize some known corresponding results in some function spaces.

How to Cite
(1)
Djeriou, A.; Heraiz, R. Some Results Concerning Localization Property of Generalized Herz, Herz-Type Besov Spaces and Herz-Type Triebel-Lizorkin Spaces. Carpathian Math. Publ. 2021, 13, 217-228.