E. Ostrovsky, L Sirota
We generalize in this short paper the classical Luzin's theorem about existence of integral on the measurable function and its multidimensional analogues on the many popular classes of rearrangement invariant (r.i.) spaces, namely, on the so-called Grand Lebesgue Spaces.
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Multiple weight Riesz and Fourier transforms in bilateral anisotropic Grand Lebesgue spaces
本文引入了各向异性（权）大Lebesgue-Riesz空间（更确切地说，是大Lebesgue-Riesz空间），它是经典Lebesgue-Riesz空间和一般的大Lebesgue空间的推广，并研究了权里斯位势和权傅里叶变换在这类空间中的有界性.
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.
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。
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。
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 derive the non-improvable Grand Lebesgue Space norm estimations for multivariate and multidimensional operator of infimal convolution.
We study in this short preprint the theory of trigonometric approximation in the so-called Banach functional rearrangement invariant Sobolev-Grand Lebesgue Spaces.
We discuss Lebesgue spaces $\mathcal{L}^p([a,b],E)$ of Lusin measurable vector-valued functions and the corresponding vector spaces $AC_{L^p}([a,b],E)$ of absolutely continuous functions. These can be used to construct Lie groups $AC_{L^p}([a,b],G)$ of absolutely continuous functions with values in an infinite-dimensional Lie group $G$. We extend the notion of $L^p$-regularity of infinite-dimensional Lie groups introduced by Glöckner to this setting and adopt several results and tools.
We generalize in this article the classical Sobolev's and Sobolev's trace inequalities on the Grand Lebesgue Spaces under monomial weight instead the classical Lebesgue or grand Lebesgue Spaces. We will distinguish the classical Sobolev's inequality and the so-called trace Sobolev's inequality.
Lusin's Theorem and Bochner Integration
它表明，逼近函数用于定义的Bochner积分可以形成使用几何漂亮的集，如球，从微分基础。此外，该形式的每个适当的和将在积分的预先指定的$ε$内，局部误差的和也小于$ε$。所有这一切都源于勒贝格点的普遍存在，勒贝格点是卢辛定理的一个结果，在讨论中包括了一个简单的证明。
We consider several fundamental properties of grand variable exponent Lebesgue spaces. Moreover, we discuss Ergodic theorems in these spaces whenever the exponent is invariant under the transformation.
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.
We describe in this short article the associate and dual (conjugate) spaces to the Grand Lebesgue Spaces by means of its embedding to the suitable exponential Orlicz ones.
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.
We investigate in this short article the fundamental function for the so-called Grand Lebesgue Spaces (GLS) and show in particular a one-to-one and mutually continuous accordance between its fundamental and generating function.
We prove that the Grand Lebesgue Space, builded on a unimodular locally compact topological group, forms a Banach algebra relative to the convolution.
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.
Compact Sets in Bide - Side Grand Lebesgue Spaces, with Applications
本文给出了双边大Lebesgue空间中的集合是紧集的几个充分条件。我们考虑应用到数值方法和基础问题。
Module of continuity for the functions belonging to the Sobolev-Grand Lebesgue Spaces
本文将经典Lebesgue空间的函数的连续模的Sobolev不等式推广到（双边）Grand Lebesgue空间上。为了证明所得结果的正确性，我们还构造了一些例子。
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。
Poincaré Inequalities in Bilateral Grand Lebesgue Spaces
本文在所谓的双边大Lebesgue空间中得到了函数及其梯度的Poincare型非渐近估计。我们也给出了一些例子来说明这些不等式的尖锐性。
We derive in this preprint the exact up to multiplicative constant non-asymptotical estimates for the norms of some non-linear in general case operators, for example, the so-called maximal functional operators, in two probabilistic rearrangement invariant norm: exponential Orlicz and Grand Lebesgue Spaces. We will use also the theory of the so-called Grand Lebesgue Spaces (GLS) of measurable functions.
We give in this short paper the exact value for norms of two operators of Hardy-Sobolev type acting between two weight Grand Lebesgue Space (GLS) based on the whole multidimensional Euclidean space.
In this paper we obtain the non-asymptotic inequalities of Poincare type between function and its weak gradient belonging the so-called Bilateral Grand Lebesgue Spaces over general metric measurable space. We also prove the sharpness of these inequalities.
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。
Riesz's and Bessel's Operators in Bilateral Grand Lebesgue Spaces
本文给出了双边大Lebesgue空间中Riesz和Bessel位势积分算子的非渐近估计。我们还举例说明了这些不等式的尖锐性。
We introduce and evaluate the degree of convexity of an unit ball, so-called, characteristic of convexity (COC) for the Grand Lebesgue Spaces, (GLS), which is a slight analog of the classical notion of the modulus of convexity (MOC).
We derive in this article the exact norm in the Grand Lebesgue Spaces (GLS) estimates for Fourier transform acting on the functions defined in the infinite local compact Abelian (LCA) group, compact or discrete.
Weighted Hardy-Littlewood average operators on bilateral grand Lebesgue spaces
本文在所谓的Bilateral Grand Lebesgue空间中得到了（广义）加权Hardy-Littlewood平均积分算子范数的非渐近精确估计.我们还举例说明了这些不等式的尖锐性。
Homogeneous kernel integral operators in Grand Lebesgue Spaces
本文研究作用于两个 Grand Lebesgue 空间之间、带齐次核的多重线性积分算子，给出其算子范数的估计与精确计算。
We give in this short paper a sharp estimate for the norm of a multivariate dilation operator generated by multi-matrix (tensor) linear argument transformation (dilation operator) between two different weight Lebesgue-Riesz and Grand Lebesgue Spaces(GLS). We consider also some examples and study the compactness of these operators.
Let $G$ a locally compact abelian group with Haar measure $μ$ and let $1<p<\infty. $ In the present paper we determine necessary and sufficient conditions on $G$ for the grand Lebesgue space $ L^{p),θ}(G)$ to be a Banach algebra under convolution.Later we characterize the multipliers of the grand Lebesgues, $L^{p,)θ}(G)$ and the small Lebesgue spaces $L^{(P'θ}$, where $\frac{1}{p}+\frac{1}{p'}=1$
We deduce an extension theorem for the so-called Sobolev-Grand Lebesgue Spaces defined on the suitable subsets of the whole finite-dimensional Euclidean space, and estimate the norms of correspondent extension operator, which may be choosed as linear.
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.
We calculate in this paper the norm of composition, multiplicative and product operator, generated by multiplicative and measurable argument transformation between two different ordinary Lebesgue-Riesz and Grand Lebesgue spaces. We set ourselves the aim to obtain the exact expression for the norm of the considered operators by means of building of appropriate examples.
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 introduce a new class of quasi-Banach spaces as an extension of the classical Grand Lebesgue Spaces for small values of the parameter, and we investigate some its properties, in particular, completeness, fundamental function, operators estimates, Boyd indices, contraction principle, tail behavior, dual space, generalized triangle and quadrilateral constants and inequalities.
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。
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.
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 establish the bilateral exact reciprocal interrelations between a tail behavior of a measurable functions and its norm in the suitable Grand Lebesgue Space (GLS) as well as Orlicz one, builded over the set with infinite measure. We bring also some examples in order to illustrate the exactness of offered estimates.
We extend the classical Lyapunov inequality on the measurable space with infinite measure and on the so-called Grand Lebesgue spaces (GLS). We find also the exact value for correspondent constant. Possible applications: Functional Analysis (for instance, interpolation of operators), Integral Equations, Probability Theory and Statistics (tail estimations for random variables) etc.
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.
Fourier Series and Transforms in Grand Lebesgue Spaces As an particular case - exponential Orlicz spaces
本文研究了定义在[-pi，pi]^ d或R^d上且属于（双边）Grand Lebesgue空间的函数的傅里叶级数和变换.作为一种特殊情况，我们得到了所谓指数型Orlicz空间中Fourier变换的一些结果。我们还构造了几个例子来证明所提供的估计的准确性。
In this paper, we show that the interpolation spaces between Grand, small or classical Lebesgue are so called Lorentz-Zygmund spaces or more generally $GΓ$-spaces. As a direct consequence of our results any Lorentz-Zygmund space $L^{a,r}({\rm Log}\, L)^β$, is an interpolation space in the sense of Peetre between either two Grand Lebesgue spaces or between two small spaces provided that $ 1<a<\infty, β\not= 0$. The method consists in computing the so called K-functional of the interpolation space and in identifying the associated norm.
We present a simple proof of the continuity, in the sense distributions, of the minors of the differential matrices of mappings belonging to grand Sobolev spaces. Such function spaces were introduced in connection with a problem on minimal integrability of the Jacobian and are useful in certain aspects of geometric function theory and partial differential equations.
In this note some structural properties of grand variable exponent Lebesgue/ Morrey spaces over spaces of homogeneous type are obtained. In particular, it is proved that the closure of the class of bounded functions and the closure of classical Morrey space in grand variable exponent Morrey space coincide if the measure of the underlying space is finite. Moreover, we get two different characterizations of this class. Further, duality and preduality of grand variable exponent Lebesghue space defined on quasi-metric measure spaces with sigma-finite measure are obtained, which is new even when un…
We intend to derive the moment and exponential tail estimates for the so-called bivariate or more generally multivariate functional operations, not necessary to be linear or even multilinear. We will show also the strong or at last weak (i.e. up to multiplicative constant) exactness of obtained estimates.
Abstract Measure Grand Lebesgue Spaces and Applications
介绍具有球基结构的抽象测度大勒贝格空间，研究其基本性质及\(\mathfrak{BO}\)算子行为，利用稀疏控制和范数不等式得出算子范数估计，并证明\(\mathfrak{BO}\)算子包含多种算子。
Estimates for tail functions under Riesz transforms in Grand Lebesgue Spaces
研究可测函数在广义Riesz型算子下的尾行为，通过L^p范数增长与Young-Fenchel变换的关系，推导出显式尾估计，并应用到经典Riesz变换中，展示算子L^p增长与输入函数内在尾行为的相互作用。
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.
Oscillating Operators in Bilateral Grand Lebesgue Spaces
本文得到了所谓双边大Lebesgue空间中振荡积分算子的非渐近估计。我们还举例说明了这些不等式的尖锐性。
We derive bilateral estimates for the constants appearing in the Fourier transform restricted theorems on the Euclidean sphere for the ordinary and especially radial functions belonging to the Lebesgue-Riesz spaces as well as belonging to the Grand Lebesgue Spaces. We obtain an exact estimate for the norm of the restriction Fourier transform operator acting on the radial functions.
Exact constant in Sobolev's and Sobolev's trace inequalities for Grand Lebesgue Spaces
本文将经典的Sobolev和Sobolev迹不等式推广到大Lebesgue空间而不是经典的Lebesgue空间上。我们将区分经典的Sobolev不等式和所谓的迹Sobolev不等式。为了简单起见，我们只考虑整个空间的情况。
Lusin Area Function and Molecular Characterizations of Musielak-Orlicz Hardy Spaces and Their Applications
Musielak-Orlicz Hardy空间的Lusin面积函数和分子特征及其应用设$φ: \mathbb R^n\times [0,\infty)\to[0,\infty)$是一个增长函数，使得$φ(x,\cdot)$是非减的，当$t>0$，$\lim_{t\to\infty}φ(x,t)=\infty$，$φ(\cdot,t)$是一个Muckenhoupt $A_\infty(\mathbb{R}^n)$在$t$中一致加权时，$φ(x,0)=0$，$φ(x,t)>0$.利用极大函数建立了Luong Dang Ky引入的Musielak-Orlicz Hardy空间$H_φ(\mathbb{R}^n)$的Lusin面积函数和分子特征.作为应用，我们得到了Musielak-Orlicz ${\mathop\mathrm{BMO}}$型空间$\mathop\mathrm{BMO}_φ(\mathbb{R}^n)$的$φ$-Carleson测度刻画，该空间已被Luong Dang Ky证明是$H_φ(\mathbb{R}^n)$的对偶空间.
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.
Non-Euclidean unification of isoperimetric profiles and grand Lebesgue-Sobolev scales
本文在完备可分度量测度空间上，通过显式变换统一等周轮廓和Grand Lebesgue-Sobolev尺度，建立嵌入定理及其逆定理，并在Heisenberg群等非欧几里得例子中验证。
Bilateral Small Lebesgue Spaces
本文研究了所谓的双边小Lebesgue空间：证明了它与大Lebesgue空间相联系，计算了它的基本函数和Boyd指数，找到了它的对偶空间等。
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.
Factorizable convergence of random variables in Grand Lebesgue Spaces
本文研究随机变量几乎处处收敛的因子分解，将经典 Lebesgue-Riesz 空间中的结果推广至 Grand Lebesgue 空间，并给出参数的精确估计、实例及本质最优性证明。
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