Bilateral Small Lebesgue Spaces
Eugene Ostrovsky, Leonid Sirota
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。
In this article we investigate the so-called Bilateral Small Lebesgue Spaces: prove that they are associated to the Grand Lebesgue spaces, calculate its fundamental functions and Boyd's indices find its dual spaces etc.
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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。
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。
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。
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。
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.
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 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.
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。
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。
Multidimensional Dilation Operators, Boyd and Shimogaki Indices of Bilateral Weight Grand Lebesque Spaces
本文计算了Bilateral Grand Lebesgue空间中伸缩算子的范数、多维Boyd指数和Shimogaki指数，并讨论了它们的一些应用。
Boundedness of Operators in Bilateral Grand Bebesgue Spaces with Exact and Weakly Exact Constant Calculation
本文研究了（双边）Grand Lebesgue空间（GLS）中某些算子（不一定是线性或次线性的）的作用，特别是双权Fourier算子、极大算子、嵌入算子等，并试图计算相应嵌入常数的精确值或至少弱精确值.我们还得到了GLS空间的内插定理，并构造了几个例子来证明所给估计的正确性。在最后两节中，我们引入了各向异性的大勒贝格空间，得到了Fourier双权不等式的一些估计，并计算了该空间的Boyd多维指标。
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。
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.
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。
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.
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 $\frac{1}{p}+\frac{1}{p'}=1$
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 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.
Let $G$ be a locally compact abelian metric group with Haar measure $λ$ and $\hat{G}$ its dual with Haar measure $μ,$ and $λ( G) $ is finite. Assume that$~1<p_{i}<\infty $, $p_{i}^{\prime }=\frac{ p_{i}}{p_{i}-1}$, $( i=1,2,3) $ and $θ\geq 0$. Let $ L^{(p_{i}^{\prime },θ}( G) ,$ $( i=1,2,3) $ be small Lebesgue spaces. A bounded measurable function $m( ξ,η) $ defined on $\hat{G}\times \hat{G}$ is said to be a bilinear multiplier on $G$ of type $[ (p_{1}^{\prime };(p_{2}^{\prime };(p_{3}^{\prime }] _{θ}$ if the bilinear operator $B_{m}$ associated with the symbol $m$, \begin{equation} B_{m}(f,g)…
We study unboundedness properties of functions belonging 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 $p_{i}$ is the smallest, is crucial. More precisely, the growth envelope is given by $E_G(L_{\vec{p}}(Ω)) = (t^{-1/\min\{p_{1}, \ldots, p_{d} \}},\min\{p_{1}, \ldots, p_{d} \})$ for mixed Lebesgue and $E_G(L_{\vec{p},q}(Ω)) = (t^{-1/\min\{p_{1}, \l…
Weighted Multilinear Square Function Bounds
本文研究了与多重线性算子相关的Littlewood-Paley-Stein平方函数的有界性.我们证明了加权Lebesgue空间边界的平方函数放松的正则性和取消条件是独立的重量，这是一个新的结果，即使在线性的情况下。对一类多线性卷积算子，证明了加权Lebesgue空间有界的充要条件.利用外推理论，对指数小于1的Lebesgue空间在多重线性环境下进行了加权上界的推广。
On the spaceability of the set of functions in the Lebesgue space $L_p$ which are in no other $L_q$
本文完全解决了Lebesgue空间$L_p[0,1]$中不属于任何更高阶$L_q$空间的函数集合的空间性刻画问题，证明其$(\alpha,\mathfrak{c})$-空间性当且仅当$\alpha<\aleph_0$，从而给出了该悬而未决问题的最终答案。
Small eigenvalues of Toeplitz operators, Lebesgue envelopes and Mabuchi geometry
本文研究极化复射影流形上符号支撑在真子集上的Toeplitz算子，证明其存在指数衰减的小特征值，并揭示特征值对数分布与Mabuchi测地线及Lebesgue包络的联系，同时将结果推广到Toeplitz矩阵。
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。
Multiplication operators on amalgam of $l^{q), θ}$ and $L^{p}$
本文定义了广义融合勒贝格函数空间和小勒贝格序列空间，研究其完备性等结构性质，证明霍尔德不等式，并探讨乘法算子。
We identify simple universal properties that uniquely characterize the Lebesgue $L^p$ spaces. There are two main theorems. The first states that the Banach space $L^p[0, 1]$, equipped with a small amount of extra structure, is initial as such. The second states that the $L^p$ functor on finite measure spaces, again with some extra structure, is also initial as such. In both cases, the universal characterization of the integrable functions produces a unique characterization of integration. Using the universal properties, we develop some of the basic elements of integration theory. We also state…
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 global well-posedness for non-elliptic derivative Schrödinger equations with small rough data
我们证明了调制空间$M^s_{2,1}(\mathbb{R}^n)$ for $n\ge 3$（$n= 2$）中三次（四次）非椭圆型导数薛定谔方程Cauchy问题的尖锐的整体适定性.在二维立方情形下，利用Gabor框架，得到了各向异性Lebesgue空间中Schr dinger半群的时间整体色散估计，其中包括$L^2_{x_1}L^\infty_{x_2,t}$空间中的时间整体极大函数估计.利用光滑效应估计和各向异性Lebesgue空间中的色散估计，证明了当Feichtinger-Segal代数或加权索伯列夫空间中的初值足够小时，二维三次双曲导数NLS存在唯一的整体解.
A small remark on small-dimensional normed barrelled spaces
结合Brian和Stuart的方法与经典Dvoretzky定理，证明无穷维Banach空间不含代数维数小于cov(𝒩)的桶型子空间，从而无穷维赋范桶型空间维数至少为cov(𝒩)。
Global Smooth Effects and Well-Posedness for the Derivative Nonlinear Schrödinger Equation with Small Rough Data
在各向异性Lebesgue空间中，得到了线性薛定谔方程解的整体光滑效应.应用这些估计，我们研究了广义椭圆和非椭圆导数非线性薛定谔方程（DNLS）的Cauchy问题，得到了调制空间$M^{3/2}_{2,1}(\mathbb{R}^n)$中小数据解的整体适定性.注意到$H^{\tilde{s}} \subset M^s_{2,1}$ $(\tilde{s}-s>n/2)$是一个最优包含，我们已经证明了DNLS的全局适定性与一类非常粗糙的数据。
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