In this paper, based on generalized Herz-type function spaces $\dot{K}_{q}^{p}(\theta)$ were introduced by Y. Komori and K. Matsuoka in 2009, we define Herz-type Besov spaces $\dot{K}_{q}^{p}B_{\beta }^{s}(\theta)$ and Herz-type Triebel-Lizorkin spaces $\dot{K}_{q}^{p}F_{\beta }^{s}(\theta)$, which cover the Besov spaces and the Triebel-Lizorkin spaces in the homogeneous case, where $\theta=\left\{\theta(k)\right\} _{k\in\mathbb{Z}}$ is a sequence of non-negative numbers $\theta(k)$ such that \begin{equation*} C^{-1}2^{\delta (k-j)}\leq \frac{\theta(k)}{\theta(j)} \leq C2^{\alpha (k-j)},\quad k>j, \end{equation*} for some $C\geq 1$ ($\alpha$ and $\delta $ are numbers in $\mathbb{R}$). Further, under the condition mentioned above on ${\theta }$, we prove that $\dot{K}_{q}^{p}\left({\theta }\right)$ and $\dot{K}_{q}^{p}B_{\beta }^{s}\left({\theta }\right)$ are localizable in the $\ell _{q}$-norm for $p=q$, and $\dot{K}_{q}^{p}F_{\beta }^{s}\left({\theta }\right)$ is localizable in the $\ell _{q}$-norm, i.e. there exists $\varphi \in \mathcal{D}({\mathbb{R}}^{n})$ satisfying $\sum_{k\in \mathbb{Z}^{n}}\varphi \left( x-k\right) =1$, for any $x\in \mathbb{R}^{n}$, such that \begin{equation*} \left\Vert f|E\right\Vert \approx \Big(\underset{k\in \mathbb{Z}^{n}}{\sum }\left\Vert \varphi (\cdot-k)\cdot f|E\right\Vert ^{q}\Big)^{1/q}. \end{equation*} Results presented in this paper improve and generalize some known corresponding results in some function spaces.
Let v : [0,∞) → ]0,∞) be a function such that the inequality v(ts) ≥ ct
-μ
v(s) holds for 0 < t,s ≤ 1 and some real number μ. In the generalized Triebel–Lizorkin spaces Fp,q
v
(R
n
), we will study the boundedness of the pseudo-differential operators whose symbols belong to Hörmander´s class S
1,δ
m
,0 ≤ δ < 1.
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