Boundedness of several operators on weighted Herz spaces

Komori, Yasuo; Matsuoka, Katsuo

doi:10.1155/2009/739134

Cited by 14 publications

(8 citation statements)

References 9 publications

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“…Recently, many authors considered the boundedness of operators on weighted Herz spaces with general Muckenhoupt weights. In [8], Komori and Matsuoka showed the boundedness of singular integral operators and fractional integrals on weighted Herz spaces. In [5], Guo and Jiang discussed the boundedness of commutators of singular integral operators on weighted Herz spaces, and as an application, they obtained the interior estimates on weighted Herz spaces for the solutions of some nondivergence elliptic equations.…”

Section: Introduction and Resultsmentioning

confidence: 99%

Parametric Marcinkiewicz integrals on weighted Herz Spaces

Hu¹,

Wang²

2015

Miskolc Math Notes

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Section: Introduction and Resultsmentioning

confidence: 99%

Parametric Marcinkiewicz integrals on weighted Herz Spaces

Hu¹,

Wang²

2015

Miskolc Math Notes

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“…The spacesK q p (θ) were first defined by Y. Komori and K. Matsuoka [8] and under the condition above, the authors studied the boundedness of singular integral operators and fractional integral operators on these spaces.…”

Section: Function Spacesmentioning

confidence: 99%

“…In this paper, we define Herz-type Besov spacesK p q B s β (θ) and Herz-type Triebel-Lizorkin spacesK p q F s β (θ) which covers Besov spaces and Triebel-Lizorkin spaces in the homogeneous case. Notice that these spaces based on generalized Herz-type function spacesK p q (θ) were introduced by Y. Komori and K. Matsuoka in [8]. After this, we treat and discuss the localization property of these spaces and then we compare our results with existing ones.…”

Section: Introductionmentioning

confidence: 96%

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

Djeriou

Heraiz

2021

Carpathian Math. Publ.

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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.

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“…Before stating our main results, let us first recall some definitions about the weighted Herz and weak Herz spaces. For more information about these spaces, one can see [6,8,9,11,16] and the references therein. Let B k = B(0, 2 k ) = {x ∈ R n : |x| ≤ 2 k } and C k = B k \B k−1 for any k ∈ Z. Denote χ k = χ C k for k ∈ Z, χ k = χ k if k ∈ N and χ 0 = χ B 0 , where χ E is the characteristic function of the set E. For any given weight function w on R n and 0 < q < ∞, we denote by L q w (R n ) the space of all functions f satisfying…”

Section: γγ (X)mentioning

confidence: 99%