Some results concerning localization property of generalized Herz, Herz-type Besov spaces and Herz-type Triebel-Lizorkin spaces
In this paper, based on generalized Herz-type function spaces K_q^p(θ) were introduced by Y. Komori and K. Matsuoka in 2009, we define Herz-type Besov spaces K_q^pB_β ^s(θ) and Herz-type Triebel-Lizorkin spaces K_q^pF_β ^s(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 equation* C^-12^δ (k-j)≤ θ(k)θ(j) ≤ C2^α (k-j),k>j, equation* for some C≥ 1 (α and δ are numbers in R). Further, under the condition mentioned above on θ , we prove that K_q^p(θ ) and K_q^pB_β ^s(θ ) are localizable in the _q-norm for p=q, and K_q^pF_β ^s(θ ) is localizable in the _q-norm, i.e. there exists φ D(R ^n) satisfying _kZ^n φ ( x-k) =1, for any xR^n, such that equation* f|E≈ (kZ^n φ (·-k)· f|E^q)^1/q. equation* Results presented in this paper improve and generalize some known corresponding results in some function spaces. , number=1, journal=Carpathian Mathematical Publications, author=Djeriou, A. and Heraiz, R., year=2021, month=Jun., pages=217–228