scholarly journals A Survey on Some Anisotropic Hardy-Type Function Spaces

2020 ◽  
Vol 36 (4) ◽  
pp. 373-456
Author(s):  
global sci
Keyword(s):  
Function Spaces ◽  
Type Function ◽  
Hardy Type

scholarly journals Some s-numbers of an integral operator of Hardy type in Banach function spaces

2016 ◽  
Vol 207 ◽  
pp. 76-97
Author(s):  
David Edmunds ◽  
Amiran Gogatishvili ◽  
Tengiz Kopaliani ◽  
Nino Samashvili
Keyword(s):  
Integral Operator ◽  
Function Spaces ◽  
Banach Function Spaces ◽  
Hardy Type

scholarly journals Growth Envelopes of Some Variable and Mixed Function Spaces

2022 ◽  
Vol 32 (3) ◽  
Author(s):  
Dorothee D. Haroske ◽  
Cornelia Schneider ◽  
Kristóf Szarvas
Keyword(s):  
Lorentz Space ◽  
Function Spaces ◽  
Lorentz Spaces ◽  
Lebesgue Spaces ◽  
Mixed Norms ◽  
Hardy Type Inequalities ◽  
Variable Lebesgue Spaces ◽  
Hardy Type ◽  
Growth Envelopes ◽  
Natural Way

AbstractWe study unboundedness properties of functions belonging to Lebesgue and Lorentz spaces with variable and mixed norms using growth envelopes. Our results extend the ones for the corresponding classical spaces in a natural way. In the case of spaces with mixed norms, it turns out that the unboundedness in the worst direction, i.e., in the direction where $$p_{i}$$ p i is the smallest, is crucial. More precisely, the growth envelope is given by $${\mathfrak {E}}_{{\mathsf {G}}}(L_{\overrightarrow{p}}(\varOmega )) = (t^{-1/\min \{p_{1}, \ldots , p_{d} \}},\min \{p_{1}, \ldots , p_{d} \})$$ E G ( L p → ( Ω ) ) = ( t - 1 / min { p 1 , … , p d } , min { p 1 , … , p d } ) for mixed Lebesgue and $${\mathfrak {E}}_{{\mathsf {G}}}(L_{\overrightarrow{p},q}(\varOmega )) = (t^{-1/\min \{p_{1}, \ldots , p_{d} \}},q)$$ E G ( L p → , q ( Ω ) ) = ( t - 1 / min { p 1 , … , p d } , q ) for mixed Lorentz spaces, respectively. For the variable Lebesgue spaces, we obtain $${\mathfrak {E}}_{{\mathsf {G}}}(L_{p(\cdot )}(\varOmega )) = (t^{-1/p_{-}},p_{-})$$ E G ( L p ( · ) ( Ω ) ) = ( t - 1 / p - , p - ) , where $$p_{-}$$ p - is the essential infimum of $$p(\cdot )$$ p ( · ) , subject to some further assumptions. Similarly, for the variable Lorentz space, it holds$${\mathfrak {E}}_{{\mathsf {G}}}(L_{p(\cdot ),q}(\varOmega )) = (t^{-1/p_{-}},q)$$ E G ( L p ( · ) , q ( Ω ) ) = ( t - 1 / p - , q ) . The growth envelope is used for Hardy-type inequalities and limiting embeddings. In particular, as a by-product, we determine the smallest classical Lebesgue (Lorentz) space which contains a fixed mixed or variable Lebesgue (Lorentz) space, respectively.

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

2021 ◽  
Vol 13 (1) ◽  
pp. 217-228
Author(s):  
A. Djeriou ◽  
R. Heraiz
Keyword(s):  
Besov Spaces ◽  
Function Spaces ◽  
Homogeneous Case ◽  
Negative Numbers ◽  
Type Function ◽  
Localization Property

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.

Sign in / Sign up

Export Citation Format

Share Document