Lebesgue spaces norm estimates for fractional integrals and derivatives
E. Ostrovsky, L. Sirota
研究了经典Lebesgue-Riesz空间和修正Besov空间中函数的分数次积分和导数的经典Lebesgue-Riesz范数和大Lebesgue范数的估计问题.
We study the problem estimation of classical Lebesgue-Riesz and Grand Lebesgue Norm for the fractional integrals and derivatives for the functions from the classical Lebesgue-Riesz spaces as well as from the modified Besov's spaces.
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An imbedding of fractional order Sobolev-Grand Lebesgue spaces, with constant evaluation
本文将经典的分数Lebesgue-Sobolev空间的嵌入定理推广到所谓的Grand Lebesgue空间，并给出了精确的常数估计。
Lebesgue-Riesz norm estimates for fractional Laplace transform
本文给出了分数Laplace积分变换在Lebesgue-Riesz和双边Grand Lebesgue空间中范数的双边非渐近估计.我们还举例说明了这些不等式的尖锐性。
Two-Weighted Norm Inequalities for the Double Hardy Transforms and Strong Fractional Maximal Functions in Variable Exponent Lebesgue Spaces
在变指数Lebesgue空间中建立了双Hardy变换和强分式极大函数的双权模估计。导出的条件是同时必要和充分的情况下，当右手侧空间的指数是常数。
Continuity of functions belonging to the fractional order Sobolev-Grand Lebesgue Spaces
本文将属于整数阶索博列夫空间的函数的连续模的经典索博列夫不等式推广到索博列夫双边大勒贝格空间上。在此基础上，我们导出了分数阶Orlicz-Sobolev嵌入定理，并研究了非高斯多参数随机场的矩形连续模。
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.
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。
Two-weight Norm Estimates for Multilinear Fractional Integrals in Classical Lebesgue Spaces
我们得到了多线性分数次积分和适当的极大函数的双权估计的准则。研究了多（次）线性强分数次极大算子的二权和一权问题，特别地，我们得到了保证这类算子的迹型不等式的充要条件.我们还建立了Festhiman-Stein型不等式，并得到了当权函数为乘积型时的单权准则.由此得到了乘积核多线性里斯位势算子的相应结果.
In this work we study the Riemann-Liouville fractional integral of order $α\in(0,1/p)$ as an operator from $L^p(I;X)$ into $L^{q}(I;X)$, with $1\leq q\leq p/(1-pα)$, whether $I=[t_0,t_1]$ or $I=[t_0,\infty)$ and $X$ is a Banach space. Our main result give necessary and sufficient conditions to ensure the compactness of the Riemann-Liouville fractional integral from $L^p(t_0,t_1;X)$ into $L^{q}(t_0,t_1;X)$, when $1\leq q< p/(1-pα)$.
In this paper we study the Riemann-Liouville fractional integral of order $α>0$ as a linear operator from $L^p(I,X)$ into itself, when $1\leq p\leq \infty$, $I=[t_0,t_1]$ (or $I=[t_0,\infty)$) and $X$ is a Banach space. In particular, when $I=[t_0,t_1]$, we obtain necessary and sufficient conditions to ensure its compactness. We also prove that Riemann-Liouville fractional integral defines a $C_0-$semigroup but does not defines a uniformly continuous semigroup. We close this study by presenting lower and higher bounds to the norm of this operator.
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。
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。
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.
Characterization of compactness of the commutators of bilinear fractional integral operators
利用双线性分数次积分算子和逐点乘法算子的符号的平均振荡性质，刻画了它们在Lebesgue空间乘积上的分解子的紧性.本文还研究了作用在加权Lebesgue空间的乘积上的算子的紧性。
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…
Multilinear Fractional Integral Operators with Generalized Kernels
本文研究核条件弱于Dini条件的多线性分数次积分算子，证明了其在加权Lebesgue空间和变指数Lebesgue空间上的有界性，并首次建立了相应多线性交换子的有界性结果。
Norms supporting the Lebesgue differentiation theorem
一个版本的Lebesgue微分定理，其中的$L^p$规范被替换为任何reversion-invariant规范。建立了这类范数支持Lebesgue微分定理的充要条件。特别是，洛伦兹，Orlicz和其他习惯规范的勒贝格定理持有的特点。
We approximate functions defined on smooth bounded domains by elements of the eigenspaces of the Laplacian or the Stokes operator in such a way that the approximations are bounded and converge in both Sobolev and Lebesgue spaces. We prove an abstract result referred to fractional power spaces of positive, self-adjoint, compact-inverse operators on Hilbert spaces, and then obtain our main result by using the explicit form of these fractional power spaces for the Dirichlet Laplacian and Stokes operators. As a simple application, we prove that all weak solutions of the incompressible convective B…
Generalized Cesàro operators, fractional finite differences and Gamma functions
本文对Sobolev-Lebesgue数列空间上的广义Cesàro算子的谱进行了全面的研究.其主要思想是在这些数列空间上将这些算子从属于合适的$C_0$ -半群。利用分数阶有限差分方法引入了这类数列空间，并证明了类似于经典Lebesgue数列空间的一些结构性质.为了说明分数阶差分的主要结果，我们给出了欧拉Γ函数的导数和的等式。最后，我们给出了广义Cesàro算子谱的一些图形表示。
We derive the non-improvable Grand Lebesgue Space norm estimations for multivariate and multidimensional operator of infimal convolution.
We extend the classical Hardy-Sobolev-Poincare-Wirtinger inequalities from the ordinary Lebesgue-Riesz spaces into the Grand Lebesgue ones, with exact constants evaluation.
We establish imbedding properties between Grand Lebesgue Spaces and (generalized) Lorentz-Zygmund ones. We extend some known previous results concerning imbedding theorems between Grand Lebesgue and classical Lebesgue-Riesz spaces and we show the exactness of the obtained estimates.
The Riemann-Liouville fractional integral in Bochner-Lebesgue spaces III
本文研究了特定阶数的 Riemann-Liouville 分数阶积分在 Bochner-Lebesgue 相关空间中的连续性与有界性，细化了 Hardy-Littlewood 的相关结果。
On Convolution in Variable Lebesgue Spaces and Applications to Fractional Navier_Stokes Equations
本文研究了变量Lebesgue空间中的卷积不等式，并应用于分数Navier-Stokes方程的局部和全局well-posedness分析。
Sharp value for the norm of composition and multiplicative operators between two different Grand Lebesgue spaces
本文计算了两个不同的普通Lebesgue-Riesz空间和Grand Lebesgue空间之间由乘性和可测辐角变换生成的复合算子、乘性算子和乘积算子的范数。我们的目标是通过建立适当的例子来获得所考虑的算子的范数的精确表达式。
Exact norm estimates for multivariate dilation operators between two Bilateral Weight Grand Lebesgue Spaces
本文给出了两个不同权Lebesgue-Riesz和Grand Lebesgue空间（GLS）之间的多元矩阵（张量）线性变元变换（dilation算子）所生成的多元dilation算子的范数的一个精确估计.我们还考虑了一些例子，并研究了这些运营商的紧性。
Fractional maximal operators on weighted variable Lebesgue spaces over the spaces of homogeneous type
本文在齐型空间上引入分数次型变指标权，建立分数次极大算子在加权变指标Lebesgue空间中的强型与弱型刻画，推广了多个已有经典结果。
Norm estimates in Grand Lebesgue Spaces for some operators, including magic square matrices
本文将积分算子在经典 Lebesgue-Riesz 空间间的范数估计推广至 Grand Lebesgue 空间，并以正幻方生成的有限维矩阵算子为例说明。
The structure of normal Hausdorff operators on Lebesgue spaces
我们考虑了Hausdorff算子的一个推广，并引入了这种算子的符号的概念。利用这一概念，我们描述了$\mathbb{R}^n.$上Lebesgue空间上正规广义Hausdorff算子的结构，并研究了其重要性质（如可逆性、谱、范数和紧性）。
We study moment rearrangement invariant spaces, which contain as particular cases the generalized Grand Lebesgue Spaces, and provide norm estimates for some operators, not necessarily linear, acting between some measurable rearrangement invariant spaces. The estimations are formulated in the terms of fundamental functions for these spaces.
We derive a sharp Grand Lebesgue Space norm estimations for normalized eigen functions for the Laplace-Beltrami operator defined on the compact smooth Riemann manifold. These estimates allow us to deduce in particular the exponential decreasing tail of distribution for these eigen functions.
Sharp estimates for module of continuity of fractional integrals and derivatives
在经典的Lebesgue-Riesz空间中，我们得到了函数的分数次积分和导数的连续模的双边估计。
The Amalgamation Property in Classical Lebesgue-Riesz Spaces, Banach Spaces with Transitive and Almost Transitive Norms and Projection Constants
本文证明了Lebesgue-Riesz空间Lp所生成的有限等价类具有合并性质当且仅当p不等于强大于2的偶数自然数.
Fractional, Maximal and Singular Operators in Variable Exponent Lorentz Spaces
引入了变指数的Lorentz空间$\mathcal{L}^{p(\cdot), q(\cdot)}$，证明了$p(t),q(t)$中的奇异积分和分数阶算子以及相应的遍历算子在这些空间中的有界性.本文的主要目的是证明这些算子在空间$\mathcal{L}^{p(\cdot), q(\cdot)}$中的有界性是可能的，而不需要指数上的局部对数条件，这对于变指数Lebesgue空间是典型的;相反，指数$p(s)$和$q(s)$应该只满足$s\to 0$和$s\to\infty$的对数型衰减条件。为了证明这一点，我们的基础上的问题的有效性的Hardy不等式在变指数Lebesgue空间的最新进展。
Grand Lebesgue Spaces norm estimates for multivariate functional operations
我们打算得到所谓的二元或更一般的多元函数运算的矩和指数尾估计，不一定是线性的，甚至是多线性的。我们还将显示强或弱（即乘法常数）获得的估计的精确性。
Besov's Type Embedding Theorem for Bilateral Grand Lebesgue Spaces
本文得到了不同BGLS中函数范数之间的Besov型非渐近范数估计。我们也给出了一些例子来说明这些不等式的尖锐性。
Mutually non isomorphic mixed-norm Lebesgue spaces
本文证明了混合范数勒贝格空间在特定条件下互不等同，仅当参数满足特定关系时才存在等同性。
Maximal and other operators in exponential Orlicz and Grand Lebesgue Spaces
我们在这个预印本中得出的确切乘法常数非渐近估计的一些非线性一般情况下的运营商，例如，所谓的最大功能运营商的规范，在两个概率重排不变的规范：指数Orlicz和大勒贝格空间。我们还将使用所谓的可测函数的大勒贝格空间（GLS）理论。
A transference result for Lebesgue spaces with $A_{\infty }$ weights and its applications
本文建立了带$A_{\infty}$权的Lebesgue空间的传递定理，将一致范数不等式推广至加权情形，简化了加权范数不等式及分数差分算子的应用，并提供了有限阶积分函数逼近不等式的新证明。
Cpmposed Grand Lebesgue Spaces
本文引入并研究了一类新的重排不变量（r.i.）Banach函数空间，即所谓的组合大勒贝格空间（CGLS），特别是积分大勒贝格空间（IGLS），它是已知的大勒贝格空间（GLS）的一些推广。我们考虑了CGLS的基本函数，计算了它的Boyd指数，得到了CGLS空间中某些（正则和奇异）算子的范数有界性，研究了共轭和相伴空间，证明了CGLS服从绝对连续范数性质等.
On the description of multidimensional normal Hausdorff operators on Lebesgue spaces
本文的主要目的是研究$\mathbb{R}^n$上正规Hausdorff算子的结构。证明了$L^2(\mathbb{R}^n)$空间中的正规Hausdorff算子与$L^2(\mathbb{R}^n; \mathbb{C}^{2^n}).$空间中的矩阵函数（其矩阵符号）相乘的算子是酉等价的。特别地，这类算子的范数和谱用符号来描述。
We introduce and study fractional variable exponents Sobolev trace spaces on any open set in the Euclidean space equipped with the Lebesgue measure. We show that every equivalence class of Sobolev functions has a quasicontinuous representatives. We use the relative capacity to characterize completely the zero trace fractional variable exponents Sobolev spaces. We also give a relative capacity criterium for removable sets.
In this paper we consider a norm based on the infinitesimal generator of the shift semigroup in a direction. The relevance of such a focus is guaranteed by an abstract representation of a fractional integro-differential operator by means of a composition of the corresponding infinitesimal generator. The main result of the paper is a theorem establishing equivalence of norms in functional spaces. Even without mentioning the relevance of this result for the constructed theory, we claim it deserves to be considered itself.
Criteria for the Boundedness of Potential Operators in Grand Lebesgue Spaces
证明了具有参数$α$，$0<α<1$的分数次积分算子在$θ_2 < (1+αq)θ_1$的广义大Lebesgue空间$L^{p), θ_1}$和$L^{q), θ_2}$之间没有界，其中$1<p<1/α$和$q=\frac{p}{1-αp}$ .此外，还证明了区间$[0,1]$上Riesz势算子$I_α$的单权不等式$$ \|I_α(fw^α)\|_{L_{w}^{q),θ(1+αq)}}\leq c\|f\|_{L_{w}^{p),θ}}, $$成立当且仅当$w\in A_{1+q/p'}$成立.
Product operators on mixed norm spaces
建立了混合范数Lebesgue空间和置换混合范数Lebesgue空间上乘积算子的不等式。它们只依赖于因子的不等式和所涉及的勒贝格指数。给出了二元拉普拉斯变换的不等式来说明该方法。在混合范数空间中给出了一个$n$ -变元Young不等式的初等证明.
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。
A note on the fractional perimeter and interpolation
我们提出的分数周长作为一个集函数插值之间的勒贝格测度和周长的意义上的De Giorgi。我们的动机来自于一个新的分数Boxing不等式，它将分数周长和Hausdorff内容联系起来，并包含了几个已知的涉及$0 < α< 1$阶索伯列夫空间$W^{α, 1}$的Gagliardo维数的不等式。
Semi-continuous convolutions on weakly periodic Lebesgue spaces
我们推导出混合拟范数估计的Lebesgue型的半连续卷积序列和功能，可以是周期性的或具有较弱的形式的周期性在某些方向上。在这些方向上，Lebesgue准范数被应用于周期而不是整个轴。
Asymptotically Weyl almost periodic functions in Lebesgue spaces with variable exponents
在变指数Lebesgue空间$L^{p(x)}.$中引入并分析了几种不同的Weyl殆周期函数和Weyl遍历分支的概念，研究了变指数Weyl概周期函数在卷积积作用下的不变性，并给出了在巴拿赫空间中抽象分数微分包含的一些例子.所引入的广义（渐近）Weyl殆周期函数类即使在函数$p(x)$具有常数值$p\geq 1,$的情况下也是新的，只要我们所考虑的函数$ϕ(x)$和$F(l,t)$满足$ϕ(x)\neq x$或$F(l,t)\neq l^{(-1)/p}.$
Weak And Strong Boundedness For P-adic Fractional Hausdorff Operator And Its Commutator
本文得到了Hausdorff算子在弱中心Morrey空间上的有界性.在此基础上，我们研究了加权p-adic弱Lebesgue空间上的p-adic分数阶Hausdorff算子的弱界.通过从Lipschitz空间中取符号函数，得到了p-adic分数Hausdorff算子为非线性算子的充分条件。此外，还得到了分数阶Hausdorff算子及其交换子在加权p-adic Lorentz空间上的强型估计。
Interpolation between H\" older and Lebesgue spaces with applications
当$X$，$Y$，$Z$是Lebesgue空间时，$\|u\|_{X}\leq C\|u\|_{Y}^θ\|u\|_{Z}^{1-θ}$型插值不等式是已知的.本文证明了用适当的H\”older半范数代替范数$\|\cdot\|_{Y}$或$\|\cdot\|_{X}$可以推广这一结果.我们甚至将证明涉及弱洛伦兹范数的更尖锐的版本。我们应用这个结果证明了Gagliardo-Nirenberg不等式的参数范围更广。
Integral Operators in Bilateral Grand Lebesgue Spaces
本文给出了从一个双边大勒贝格空间（BGLS）作用到另一个双边大勒贝格空间的算子的范数估计。并举例说明所给不等式的尖锐性。
Composition operators between two different bilateral grand Lebesgue spaces
本文考虑由两个不同的Grand Lebesgue空间（GLS）之间的非奇异可测变换生成的复合算子，研究了复合算子的有界性、紧性和本质范数.
The Weighted $\boldsymbol{L}^{\boldsymbol{p}}$ estimates for the fractional Hardy operator and a class of integral operators on the Heisenberg group
研究海森堡群上分数阶哈代算子等的加权$\boldsymbol{L}^{\boldsymbol{p}}$估计，通过研究$n$维分数阶哈代算子弱估计、$m$线性积分算子界等，得到哈代等算子在加权勒贝格空间的精确界及豪斯多夫算子在加权$L^p$空间的估计。
Linear bound for the dyadic paraproduct on weighted Lebesgue space $L_2(w)$
并元仿积在加权Lebesgue空间$L_p(w)$中有界当且仅当权$w$属于Muckenhoupt类$A_p^d$ .然而，即使在最简单的$L_2(w)$的情况下，并元仿积的范数的急剧界是未知的。本文利用Bellman函数技巧证明了加权Lebesgue空间$L_2(w)$中并元仿积范数的线性界，并将此结果推广到$L_p(w)$的情形.
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.
Cesaro-Hardy operators on bilateral grand Lebesgue spaces
本文在所谓的Bilateral Grand Lebesgue空间中得到了（广义）Cesaro-Hardy积分算子范数的非渐近估计.我们还举例说明了这些不等式的尖锐性。
Bounded operators on mixed norm Lebesgue spaces
本文研究了混合范数Lebesgue空间上的两类有界算子，即复合算子和乘积算子。给出了混合范数Lebesgue空间上有界复合算子的完整刻画。对某类积分算子，我们给出了有界性的充分条件。最后，我们将开发的技术应用到调查的Hardy-Steklov型运营商。
Sharp Weight Hardy-Sobolev inequalities for Grand Lebesgue Spaces
本文给出了在整个多维欧几里得空间上作用于两个权大Lebesgue空间（GLS）之间的两个Hardy-Sobolev型算子范数的精确值.
Extensions of Rubio de Francia's extrapolation theorem in variable Lebesgue space and application
在变指数Lebesgue空间上得到了Rubio de Fracia外推定理的一个变形.作为结果，我们得到了强奇异积分算子、具有粗糙核的奇异积分算子、与球面平均相关的分数次极大算子、Bochner-Riesz算子在变Lebesgue空间中有界的条件.
Fourier transform acting on the functions defined in the infinite LCA groups. A Grand Lebesgue Spaces approach
本文给出了作用在有限局部紧阿贝尔群（LCA）上的函数的傅里叶变换在大勒贝格空间（GLS）中的精确范数估计.
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