Douadi Drihem
In this paper, we investigate the inequality \begin{equation*} \left\Vert f(\cdot +h)\right\Vert_{p\left( \cdot \right) }\leq A\left\Vert f\right\Vert_{p\left( \cdot \right) },\quad h\in \mathbb{R}^{n}, A>0 \end{equation*} under some suitable assumptions on the function $f$ and the variable exponent $p$.
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In this paper, the $L^p_v(\R)$-boundedness of the Dunkl-Hausdorff operator $\displaystyle H_{\al,ϕ} f(x)=\ent\frac{ |ϕ(t)|}{|t|^{2\al+2}}f\lf(\frac{x}{t}\rh) dt $ has been characterized and for a certain type of weight $v$, the precise value of the norm $\|H_{\al,ϕ}\|_{L^p_v(\R)\to L^p_v(\R)}$ has been obtained. This covers several of the existing results. Analogous results in two dimensions have also been proved.
On the boundedness of periodic Fourier integral operators in Lebesgue spaces with variable exponent
本文研究了n维环面上变指数Lebesgue空间中周期Fourier积分算子的有界性，针对象征属于Hörmander类的$(\rho,\delta)$型算子，其中$0\leq\delta<\rho\leq1$。
Boundedness of Hausdorff-type operators with two-variable kernels on Lebesgue spaces
该研究提出涵盖各类Hausdorff算子的广义Hausdorff型算子概念（核含外部变量、作用于不同集合），并证明了其在Lebesgue空间$L^p$上有界性的经典结果的推广与类似结论，拓展了相关算子理论。
In this paper, we introduce the Hausdorff operator associated with the Opdam--Cherednik transform and study the boundedness of this operator in various Lebesgue spaces. In particular, we prove the boundedness of the Hausdorff operator in Lebesgue spaces, in grand Lebesgue spaces, and in quasi-Banach spaces that are associated with the Opdam--Cherednik transform. Also, we give necessary and sufficient conditions for the boundedness of the Hausdorff operator in these spaces.
It is shown that that the fractional integral operators with the parameter $α$, $0<α<1$, are not bounded between the generalized grand Lebesgue spaces $L^{p), θ_1}$ and $L^{q), θ_2}$ for $θ_2 < (1+αq)θ_1$, where $1<p<1/α$ and $q=\frac{p}{1-αp}$. Besides this, it is proved that the one--weight inequality $$ \|I_α(fw^α)\|_{L_{w}^{q),θ(1+αq)}}\leq c\|f\|_{L_{w}^{p),θ}}, $$ where $I_α$ is the Riesz potential operator on the interval $[0,1]$, holds if and only if $w\in A_{1+q/p'}$.
In this paper, we introduced the local and global mixed Morrey-type spaces, and some properties of these spaces are also studied. After that, the necessary conditions of the boundedness of fractional integral operators $I_α$ are studied respectively in mixed-norm Lebesgue spaces and the local mixed Morrey-type spaces. Last but not least, the boundedness of $I_α$ is obtained by Hardy operators' boundedness in weighted Lebesgue spaces. As corollaries, we acquire the boundedness of $I_α$ in mixed Morrey spaces(\hyperlink{Corollary 5.6}{Corollary 5.6}) and mixed Lebesgue spaces(\hyperlink{Corollar…
In this paper, the sharp maximal theorem is generalized to mixed-norm ball Banach function spaces, which is defined as Definition 2.7. As an application, we give a characterization of BMO via the boundedness of commutators of fractional integral operators on mixed-norm Lebesgue spaces. Moreover, the characterization of homogeneous Lipschitz space is also given by the boundedness of commutators of fractional integral operators on mixed-norm Lebesgue spaces. Finally, two applications of Corollary 6.4 are given.
Boundedness of Erdélyi--Kober Integrals and Mellin Fractional Integrals on Weighted Lebesgue Spaces
本文研究正实轴上加权Lebesgue空间中Erdélyi–Kober与Mellin分数积分的有界性，基于加权Hardy不等式等方法推导了相关权函数条件，揭示两类算子的联系并推广了已有结果。
In this paper the boundedness of the weighted iterated Hardy-type operators $T_{u,b}$ and $T_{u,b}^*$ involving suprema from weighted Lebesgue space $L_p(v)$ into weighted Cesàro function spaces ${\operatorname{Ces}}_{q}(w,a)$ are characterized. These results allow us to obtain the characterization of the boundedness of the supremal operator $R_u$ from $L^p(v)$ into ${\operatorname{Ces}}_{q}(w,a)$ on the cone of monotone non-increasing functions. For the convenience of the reader, we formulate the statement on the boundedness of the weighted Hardy operator $P_{u,b }$ from $L^p(v)$ into ${\oper…
In this paper, we establish the sharp boundedness of p-adic multilinear Hausdorff operators on the product of Lebesgue and central Morrey spaces associated with both power weights and Muckenhoupt weights. Moreover, the boundedness for the commutators of p-adic multilinear Hausdorff operators on the such spaces with symbols in central BMO space is also obtained.
Recently V. Kokilashvili, N. Samko, and S. Samko have proved a sufficient condition for the boundedness of the Cauchy singular integral operator on variable Lebesgue spaces with radial oscillating weights over Carleson curves. This condition is formulated in terms of Matuszewska-Orlicz indices of weights. We prove a partial converse of their result.
For the Hardy-Littlewood maximal and Calderón-Zygmund operators, the weighted boundedness on the Lebesgue spaces are well known. We extend these to the Orlicz-Morrey spaces. Moreover, we prove the weighted boundedness on the weak Orlicz-Morrey spaces. To do this we show the weak-weak modular inequality. The Orlicz-Morrey space and its weak version contain weighted Orlicz, Morrey and Lebesgue spaces and their weak versions as special cases. Then we also get the boundedness for these function spaces as corollaries.
In this paper we consider composition operator generated by nonsingular measurable transformation between two different Grand Lebesgue Spaces (GLS); we investigate the boundedness, compactness and essential norm of composition operators.
In this short article we introduce so-called anisotropic (weight) Grand Lebesgue Spaces (more exactly, Grand Lebesgue-Riesz Spaces), which are generalization of the classical Lebesgue-Riesz Spaces and ordinary Grand Lebesgue Spaces, and investigate the boundedness of weight Riesz potential and weight Fourier transforms in this spaces.
Let $L:=-Δ+V$ be the Schrödinger operator on $\mathbb{R}^n$ with $n\geq 3$, where $V$ is a non-negative potential which belongs to certain reverse Hölder class $RH_q(\mathbb{R}^n)$ with $q\in (n/2,\,\infty)$. In this article, the authors obtain the quantitative weighted boundedness of Littlewood--Paley functions $g_L$, $S_L$ and $g_{L,\,λ}^\ast$, associated to $L$, on weighted Lebesgue spaces $L^p(w)$, where $w$ belongs to the class of Muckenhoupt $A_p$ weights adapted to $L$.
In this note we study boundedness of a large class of maximal operators in Sobolev spaces that includes the spherical maximal operator. We also study the size of the set of Lebesgue points with respect to convergence associated with such maximal operators.
We provide the conditions for the boundedness of the Bochner-Riesz operator acting between two different Grand Lebesgue Spaces. Moreover we obtain a lower estimate for the constant appearing in the Lebesgue-Riesz norm estimation of the Bochner-Riesz operator and we investigate the convergence of the Bochner-Riesz approximation in Lebesgue-Riesz spaces.
We introduce a so-called restricted, in particular, discrete version of (Banach) Grand Lebesgue Spaces (GLS), investigate its properties and derive the conditions of coincidence with the classical ones. We show also that these spaces forms also a Banach algebra relative the convolution operation on the unimodular local compact topological group equipped with Haar's measure, alike in the complete GLS case.
We study in this short report the boundedness of classical non-linear operators: Nemytskii, Urysohn, Hammerstein acting from one Grand Lebesgue Space to another one, and deduce some its upper norm estimates. We bring also some examples to illustrate the exactness of our estimates.
In this article we investigate an action of some operators (not necessary to be linear or sublinear) in the so-called (Bilateral) Grand Lebesgue Spaces (GLS), in particular, double weight Fourier operators, maximal operators, imbedding operators etc. We intend to calculate an exact or at least weak exact values for correspondent imbedding constant. We obtain also interpolation theorems for GLS spaces.We construct several examples to show the exactness of offered estimations. In two last sections we introduce anisotropic Grand Lebesgue Spaces, obtain some estimates for Fourier two-weight inequa…
We obtain boundedness from a product of Lebesgue or Hardy spaces into Hardy spaces under suitable cancellation conditions for a large class of multilinear operators that includes the Coifman-Meyer class, sums of products of linear Calderon-Zygmund operators and combinations of these two types.
Boundedness of composition operators from Lorentz spaces to Orlicz spaces
本研究针对函数空间间复合算子有界性研究成果较少的问题，给出了从Lorentz空间到Orlicz空间的复合算子有界性的充要条件，还提供了相关反例及更多满足主结果条件的复合映射实例。
Discrete Riesz Potentials on Discrete Weighted Morrey Spaces
本文证明了离散分数阶Hardy-Littlewood极大算子和离散Riesz位势在离散加权Lebesgue及Morrey空间上的有界性，其中极大算子的结果在无权重情形下也是新的。
In this study, we investigate the boundedness of composition operators acting on Morrey spaces and weak Morrey spaces. The primary aim of this study is to investigate a necessary and sufficient condition on the boundedness of the composition operator induced by a diffeomorphism on Morrey spaces. In particular, detailed information is derived from the boundedness, i.e., the bi-Lipschitz continuity of the mapping that induces the composition operator follows from the continuity of the composition mapping. The idea of the proof is to determine the Morrey norm of the characteristic functions, and …
In this article we introduce and investigate a new class of rearrangement invariant (r.i.) Banach function spaces, so-called Composed Grand Lebesgue Spaces (CGLS), in particular, Integral Grand Lebesgue Spaces (IGLS), which are some generalizations of known Grand Lebesgue Spaces (GLS). We consider the fundamental functions of CGLS, calculate its Boyd's indices, obtain the norm boundedness some (regular and singular) operators in this spaces, investigate the conjugate and associate spaces, show that CGLS obeys the absolute continuous norm property etc.
We carry on the study of Fourier integral operators of H{ö}rmander's type acting on the spaces $(\mathcal{F}L^p)_{comp}$, $1\leq p\leq\infty$, of compactly supported distributions whose Fourier transform is in $L^p$. We show that the sharp loss of derivatives for such an operator to be bounded on these spaces is related to the rank $r$ of the Hessian of the phase $Φ(x,η)$ with respect to the space variables $x$. Indeed, we show that operators of order $m=-r|1/2-1/p|$ are bounded on $(\mathcal{F}L^p)_{comp}$, if the mapping $x\longmapsto\nabla_xΦ(x,η)$ is constant on the fibers, of codimension …
Interpolation on Convex-set Valued Lebesgue Spaces and its Applications
本文在凸集值Lebesgue空间上证明了两个插值定理，推广了经典插值定理，并据此得到了凸集值分数平均算子与极大算子的有界性及矩阵权重的逆向分解性质。
New Multilinear Littlewood--Paley $g_λ^{*}$ Function and Commutator on weighted Lebesgue Spaces
本文利用新权函数 $A_{\vec p}^{θ}(φ)$ 研究新多线性 Littlewood–Paley $g_λ^{*}$ 函数及其与新 $BMO$ 函数生成的交换子、迭代交换子，证明其在加权 Lebesgue 空间上的有界性，并推广已有线性与多线性结果。
Boundedness of commutator generated by fractional integral operator and Orlicz-BMO function
研究了分数积分算子与Orlicz-BMO函数生成的交换子从Orlicz-Hardy空间到Lebesgue空间的有界性，以及其在弱Lebesgue空间中的有界性。
We prove Rubio de Francia extrapolation results in Lebesgue and grand Lebesgue spaces for quasi monotone functions with $QB_{β,p}$ weights. The extrapolation in Lebesgue spaces with the weight class $QB_{β,\infty}$ has also been investigated. As an application, we characterize the boundedness of the Hardy averaging operator for quasi monotone functions in the grand Lebesgue spaces.
The main goal of this paper is to prove a two-weight criteria for multidimensio-nal Hardy type operator from weighted Lebesgue spaces into $p$-convex weighted Banach function spaces. Analogously problem for the dual operator is considered. As application we prove a two-weight criteria for boundedness of multidimensional geometric mean operator and sufficient condition on the weights for boundedness of certain sublinear operator from weighted Lebesgue spaces into weighted Musielak-Orlicz spaces.
Criteria for boundedness of a class of integral operators from $L_p$ to $L_q$ for $1<q<p<\infty$
针对核满足更弱条件的非负核积分算子，本文给出其在 $1<q<p<\infty$ 时从 $L_p$ 到 $L_q$ 有界的判据。
In this paper, boundedness of Hausdorff operator on weak central Morrey space is obtained. Furthermore, we investigate the weak bounds of p- adic fractional Hausdorff Operator on weighted p-adic weak Lebesgue Space. We also obtain the sufficient condition of commutators of p-adic fractional Hausdorff Operator by taking symbol function from Lipschitz space. Moreover, strong type estimates for fractional Hausdorff Operator and its commutator on weighted p-adic Lorentz space are also acquired.
In the paper two-weighted norm estimates with general weights for Hardy-type transforms, maximal functions, potentials and Calderón-Zygmund singular integrals in variable exponent Lebesgue spaces defined on quasimetric measure spaces $(X, d, μ)$ are established. In particular, we derive integral-type easily verifiable sufficient conditions governing two-weight inequalities for these operators. If exponents of Lebesgue spaces are constants, then most of the derived conditions are simultaneously necessary and sufficient for appropriate inequalities. Examples of weights governing the boundedness …
We obtain some new characterizations of a variable version of Lipschitz spaces in terms of the boundedness of commutators of sharp maximal functions, fractional maximal functions or fractional maximal commutators in the context of the variable Lebesgue spaces, where the symbols of the commutators belong to the variable Lipschitz space. A useful tool is that a symbol $b$ belongs a variable Lipschitz space of pointwise type if and only if $b$ belongs to a variable Lipschitz space of integral type.
In this note we introduce grand grand Morrey spaces, in the spirit of the grand Lebesgue spaces. We prove a kind of \textit{reduction lemma} which is applicable to a variety of operators to reduce their boundedness in grand grand Morrey spaces to the corresponding boundedness in Morrey spaces. As a result of this application, we obtain the boundedness of the Hardy-Littlewood maximal operator and Calderón-Zygmund operators in the framework of grand grand Morrey spaces.
The Riemann-Liouville fractional integral in Bochner-Lebesgue spaces IV
本文在Bochner-Lebesgue空间中研究Riemann-Liouville分数阶积分，解决 $α>1/p$ 时的有界性剩余问题，并推广到非标准函数空间。
We study the mapping property of the commutator of bilinear Hardy-Littlewood maximal operator in homogeneous Triebel-Lizorkin space. We also show that the commutator of bilinear Hardy-Littlewood maximal operator is a compact operator acting on product of Lebesgue spaces. The results are new even in the linear case.
We reduce the boundedness of operators in Morrey spaces $L_p^r({\mathbb R}^n)$, its preduals, $H^{\varrho}L_p ({\mathbb R}^n)$, and their preduals $\overset{\circ}{L}{}^r_p({\mathbb R}^n)$ to the boundedness of the appropriate operators in Lebesgue spaces, $L_p({\mathbb R}^n)$. Hereby, we need a weak condition with respect to the operators which is satisfied for a large set of classical operators of harmonic analysis including singular integral operators and the Hardy-Littlewood maximal function. The given vector-valued consideration of these issues is a key ingredient for various applications…
In the present work, we find useful and explicit necessary and sufficient conditions for linear and multilinear multiplier operators of Coifman-Meyer type, finite sum of products of Calderón-Zygmund operators, and also of intermediate types to be bounded from a product of Lebesgue or Hardy spaces into a Hardy space. These conditions state that the symbols of the multipliers $σ(ξ_1,\dots , ξ_m)$ and their derivatives vanish on the hyperplane $ξ_1+\cdots+ξ_m=0$.
Discrete Hilbert Transform And Discrete Mikhlin Multiplier On Discrete Variable Lebesgue Space
本文在变量离散Lebesgue空间上证明了离散Hilbert变换的有界性，并给出了离散Mikhlin乘子在该空间上有界的充分条件。
In this paper, the central BMO spaces with variable exponent are introduced. As an application, we characterize these spaces by the boundedness of commutators of Hardy operator and its dual operator on variable Lebesgue spaces. The boundedness of vector-valued commutators on Herz spaces with variable exponent are also considered.
$L^{\vec{p}}-L^{\vec{q}}$ Boundedness of Multiparameter Forelli-Rudin Type Operators on Tube Domains Over The Forward Light Cones
本文针对前向光锥上管域的加权Lebesgue空间，刻画了两类多参数Forelli-Rudin型算子在指定参数范围的$L^{\vec{p}}-L^{\vec{q}}$有界性充要条件，为对应Bergman投影分析提供支撑。
Boundedness of composition operator in Orlicz-Morrey spaces
研究奥利奇 - 莫雷空间上复合算子有界性的充要条件，结果涵盖特殊情形，还刻画了弱奥利奇 - 莫雷空间上复合算子有界性，该空间包含奥利奇 - 莫雷空间。
In this article, the authors establish a general (two-weight) boundedness criterion for a pair of functions, $(F,f)$, on $\mathbb{R}^n$ in the scale of weighted Lebesgue spaces, weighted Lorentz spaces, (Lorentz--)Morrey spaces, and variable Lebesgue spaces. As applications, the authors give a unified approach to prove the (two-weight) boundedness of Calderón--Zygmund operators, Littlewood--Paley $g$-functions, Lusin area functions, Littlewood--Paley $g^\ast_λ$-functions, and fractional integral operators, in the aforementioned function spaces. Moreover, via applying the above (two-weight) bou…
In this work we study boundedness of Littlewood-Paley-Stein square func- tions associated to multilinear operators. We prove weighted Lebesgue space bounds for square functions under relaxed regularity and cancellation conditions that are independent of weights, which is a new result even in the linear case. For a class of multilinear convolu- tion operators, we prove necessary and sufficient conditions for weighted Lebesgue space bounds. Using extrapolation theory, we extend weighted bounds in the multilinear setting for Lebesgue spaces with index smaller than one.
Morrey spaces can complement the boundedness properties of operators that Lebesgue spaces can not handle. Morrey spaces which we have been handling are called classical Morrey spaces. However, classical Morrey spaces are not totally enough to describe the boundedness properties. To this end, we need to generalize parameters $p$ and $q$, among others $p$.
Boundedness and norm of certain p-adic Hardy-Littlewood-Pólya-type operators
本文研究了p进Hardy-Littlewood-Pólya型算子的有界性和范数，通过参数引入，刻画了其在p进加权Lebesgue空间上的L^q-L^r有界性并给出了特殊情形下的精确范数估计。
We study two classes of bounded operators on mixed norm Lebesgue spaces, namely composition operators and product operators. A complete description of bounded composition operators on mixed norm Lebesgue spaces are given. For a certain class of integral operators, we provide sufficient conditions for boundedness. We conclude by applying the developed technique to the investigation of Hardy-Steklov type operators.
In this paper we introduce generalized grand Morrey spaces in the framework of quasimetric measure spaces, in the spirit of the so-called grand Lebesgue spaces. We prove a kind of reduction lemma which is applicable to a variety of operators to reduce their boundedness in generalized grand Morrey spaces to the corresponding boundedness in Morrey spaces, as a result of this application, we obtain the boundedness of the Hardy-Littlewood maximal operator as well as the boundedness of Calderón-Zygmund potential type operators. Boundedness of Riesz type potential operators are also obtained in the …
By the Hardy-Littlewood-Sobolev theorem the classical Riesz potential is bounded on Lebesgue spaces. E. Nakai and H. Sumitomo [16] extended that theorem to the Orlicz spaces. We introduce generalized potential operators on commutative hypergroups and under some assumptions on the kernel we showed the boundedness of these operators from Lebesgue space into certain Orlicz space. Our result is an analogue of Theorem 1.3 in [16].
We prove the $\ell^s$-boundedness of a family of integral operators with an operator-valued kernel on UMD Banach function spaces. This generalizes and simplifies earlier work by Gallarati, Veraar and the author, where the $\ell^s$-boundedness of this family of integral operators was shown on Lebesgue spaces. The proof is based on a characterization of $\ell^s$-boundedness as weighted boundedness by Rubio de Francia.
The main purpose of this paper is to investigate the behaviour of fractional integral operators associated to a measure on a metric space satisfying just a mild growth condition, namely that the measure of each ball is controlled by a fixed power of its radius. This allows, in particular, non--doubling measures. It turns out that this condition is enough to build up a theory that contains the classical results based upon the Lebesgue measure on euclidean space and their known extensions for doubling measures. We start by analyzing the images of the Lebesgue spaces associated to the measure. Th…
We obtain one variant of the extrapolation theorem of Rubio de Fracia for variable exponent Lebesgue spaces. As a consequence we obtain conditions guarantee boundedness of strongly singular integral operators, singular integral operators with rough kernels, fractional maximal operators related to spherical means, Bochner-Riesz operators in variable Lebesgue spaces.
We introduce the Lorentz space $\mathcal{L}^{p(\cdot), q(\cdot)}$ with variable exponents $p(t),q(t)$ and prove the boundedness of singular integral and fractional type operators, and corresponding ergodic operators in these spaces. The main goal of the paper is to show that the boundedness of these operators in the spaces $\mathcal{L}^{p(\cdot), q(\cdot)}$ is possible without the local log-condition on the exponents, typical for the variable exponent Lebesgue spaces; instead the exponents $p(s)$ and $q(s)$ should only satisfy decay conditions of log-type as $s\to 0$ and $s\to\infty$. To prove…
In this paper we study some estimates of norms in variable exponent Lebesgue spaces for a singular integral operators that are imaginary powers of the Laplace operator in $\R^n$. Using Mellin transform argument, from this estimates we obtain boundedness for a family of maximal operators in variable exponent Lebesgue spaces, which are closely related to the (weak) solution of the wave equation.42B25, 42B20, 46E30, 44A10, 42B10, 35L05
Boundedness of metaplectic operators within $L^p$ spaces, applications to pseudodifferential calculus, and time-frequency representations
本文推广了 Housdorff-Young 不等式，刻画了所有 metaplectic 算子在 $L^p-L^q$ 空间的有界性条件，并基于此推导了伪微分算子的有界性结果及 metaplectic Wigner 分布在调制空间中的性质。
The Generalized Grand Wiener Amalgam Spaces and the boundedness of Hardy-Littlewood maximal operators
本文将 grand Wiener 融合空间推广为基于广义 grand Lebesgue 空间的广义 grand Wiener 融合空间，研究其基本性质与嵌入关系，并讨论 Hardy-Littlewood 极大算子在相应空间之间的有界性与无界性。
Multilinear Strongly Singular Integral Operators with Generalized Kernels on RD-Spaces
本文在RD-空间上引入一类具有广义核的多线性强奇异积分算子，证明了其在加权Lebesgue空间和广义加权Morrey空间上的有界性及端点估计，推广并扩展了欧氏空间中的相关已知结论。
A new general Hormander type condition involving anisotropies and mixed norms is introduced, and boundedness results for Fourier multi- pliers on anisotropic Besov and Triebel-Lizorkin spaces of distributions with mixed Lebesgue norms are obtained. As an application, the continuity of such operators is established on mixed Sobolev and Lebesgue spaces too. Some lifting properties and equivalent norms are obtained as well.
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