On Some Discrete Inequalities in Normed Linear Spaces

赋范空间的离散子群是自由的

Discrete subgroups of normed spaces are free

本文利用初等子模型理论与奇异紧致性定理，将Ancel等人关于赋范空间可数离散子群为自由Abel群的结论推广到任意基数的离散子群。

拟范数群中双边逼近的若干精确常数

Some exact constants for bilateral approximations in quasi-normed groups

本文在拟范数阿贝尔群中利用带精确常数的双边 Bernstein-Jackson 不等式建立最佳逼近的逆定理与正定理，并通过递减重排和 E-泛函给出误差估计及数值与谱逼近实例。

Multi-normed spaces, based on non-discrete measures, and their tensor products

It was A. Lambert who discovered a new type of structures, situated, in a sense, between normed spaces and (abstract) operator spaces. His definition was based on the notion of amplification a normed space by means of spaces $\ell_2^n$. Afterwards several mathematicians investigated more general structure, "p-multi-normed space", introduced with the help of spaces $\ell_p^n$; $1\le p\le\infty$. In the present paper we pass from $\ell_p$ to $L_p(X,μ)$ with an arbitrary measure. This happened to be possible in the frame-work of the non-coordinate ("index-free") approach to the notion of amplific…

非正算子的解耦、多重线性对偶性与分解

Disentanglement, Multilinear Duality and Factorisation for non-positive operators

本文首次将多重线性对偶与分解理论从正算子推广至赋范空间上的一般线性算子，通过向量值重述、拉德马赫型几何性质及p-稳定随机变量实现解耦。

On inequalities for convex functions

We study some properties convex functions fulfill. Among the conclusions we obtain from such result, we are able to prove some nontrivial inequalities among real numbers, and we give an improvement of the reverse triangle inequality in the particular case where three aligned points are considered in a normed space.

A discrete form of the Beckman-Quarles theorem for two-dimensional strictly convex normed spaces

Let X be a real normed vector space and dim X \ge 2. Let d>0 be a fixed real number. We prove that if x,y \in X and ||x-y||/d is a rational number then there exists a finite set {x,y} \subseteq S(x,y) \subseteq X with the following property: for each strictly convex Y of dimension 2 each map from S(x,y) to Y preserving the distance d preserves the distance between x and y. It implies that each map from X to Y that preserves the distance d is an isometry.

$L^0$--convex compactness and its applications to random convex optimization and random variational inequalities

In 2010, Gordan Žitković introduced the notion of convex compactness for a convex subset of a linear topological space and gave some important applications to both nonlinear analysis and mathematical economics in [ Gordan Žitković, Convex compactness and its applications, Math. Finance Econom. 3(1) (2010) 1--12 ]. Motivated by Gordan Žitković's idea, in this paper we introduce the notion of $L^0$--convex compactness for an $L^0$--convex subset of a topological module over the topological algebra $L^0(\mathcal{F},K)$, where $L^0(\mathcal{F},K)$ is the algebra of equivalence classes of random va…

赋范空间中 $\varepsilon$-Dvoretzky 猜想的弱版本

A weak version of the $\varepsilon$-Dvoretzky conjecture for normed spaces

本文证明了赋范空间中ε-Dvoretzky猜想的弱版本，揭示了R^n中存在一个维数至少为c log n / |log ε|的子空间，使得给定范数ε-接近于1-无条件范数。

离散空间中的临界Hardy不等式（$L^p$空间）

Critical discrete Hardy inequalities in $L^p$

该研究在离散$L^p$框架下，针对半直线上离散Dirichlet Laplacian和整数上离散分数阶Laplacian，利用Sobolev-Bregman型建立了临界Hardy不等式及基态表示，明确了Hardy权的算子表达形式。

取值于半赋范空间中的绝对可积与上可积函数空间的完备性

Completeness of the space of absolutely and upper integrable functions with values in a semi-normed space

本文研究取值于半赋范空间和局部凸拓扑向量空间函数的绝对可积性，基于ρ-变差测度引入上积分方法定义上可积函数空间，证明其序列完备性及绝对可积子空间的闭性。

关于Hardy与Copson不等式的改进

On the improvements of Hardy and Copson inequalities

本文改进了一维离散Hardy不等式并证明其最优性，在特定情形下改进了Copson不等式，同时研究了相应序列空间的完备性、对偶性与可分性等基本结构。

Poincaré type inequalities on the discrete cube and in the CAR algebra

We prove Lp Poincare inequalities for functions on the discrete cube and their discrete gradient. We thus recover an exponential inequality and the concentration phenomenon for the uniform probability on the cube first obtained by Bobkov and Gotze. Inequalities involving the discrete gradient and powers of the discrete Laplacian are also considered, for the Lp norm or more general ones. Similar results hold true, replacing functions on the cube by elements of the CAR algebra and considering the annihilation operators and the number operator.

可变 $\ell^{p(\cdot)}(\mathbb{Z})$ 空间上的经典离散算子

Classical discrete operators on variable $\ell^{p(\cdot)}(\mathbb{Z})$ spaces

本研究利用离散加权范数不等式与Rubio de Francia算法，证明离散Hardy-Littlewood极大算子在对偶空间有界时，离散Hilbert变换和Riesz位势在可变ℓ^{p(·)}(ℤ)空间有界，还得到离散分数次极大算子的向量值不等式。

Hartley-Fourier 卷积算子的 $L_p$ 空间有界性及其应用

Boundedness in $L_p$ spaces for the Hartley-Fourier convolutions operator and their applications

本文研究 Hartley-Fourier 卷积算子的 $L_p$ 有界性，建立 Young 型不等式和赋范环结构，并应用于 Fredholm 型积分方程与 Barbashin 方程的可解性分析。

从Brunn-Minkowski到Prékopa-Leindler和Borell-Brascamp-Lieb：离散不等式

From Brunn-Minkowski to Prékopa-Leindler and Borell-Brascamp-Lieb: discrete inequalities

本文通过Brunn-Minkowski不等式推导出Prékopa-Leindler和Borell-Brascamp-Lieb不等式，并证明了其在整数空间上的离散形式。

通过耦合方法研究离散空间中的HWI不等式

HWI inequalities in discrete spaces via couplings

本文将高斯HWI不等式的耦合证明方法推广到离散空间，在离散超立方体和离散环面上建立新的插值不等式，并改进了Bobkov和Tetali的修正对数Sobolev不等式。

离散霍尔德与反 Hardy 型不等式在洛伦茨序列空间中

Discrete reversed Hardy inequalities and Hölder estimates in Lorentz sequence spaces

本文在洛伦茨序列空间中建立了新的最优反 Hardy 型不等式及不同范数间的估计，推导出补充函数情况的尖锐霍尔德型不等式。

Discrete Hardy-type Inequalities

This paper studies the Hardy-type inequalities on the discrete intervals. The first result is the variational formulas of the optimal constants. Using these formulas, one may obtain an approximating procedure and the known basic estimates of the optimal constants. The second result, which is the main innovation of this paper, is about the factor of basic upper estimates. An improved factor is presented, which is smaller than the known one and is best possible. Some comparison results are included for comparing the optimal constants on different intervals.

度量测度空间上的振荡不等式、下球增长与Sobolev嵌入

Oscillation inequalities, lower ball growth and Sobolev embeddings on metric measure spaces

研究度量测度空间上球函数下界、重排振荡与Sobolev型嵌入的关系，证明下界估计等价于点态对称化不等式，并导出赋范振荡不等式和Sobolev嵌入。

巴拿赫空间中的斜参数化几何常数

Skew-Parameterized Geometric Constants in Banach Spaces

该研究在Amini-Harandi等人的几何常数基础上，定义了斜参数化几何常数族，建立相关公式与条件，确定希尔伯特空间上的常数精确值，推导非方性等性质的条件，为检测巴拿赫空间相关几何性质提供统一框架。

Agmon-Kolmogorov inequalities on $\ell^2(\Bbb Z^d)$

Landau-Kolmogorov inequalities have been extensively studied on both continuous and discrete domains for an entire century. However, the research is limited to the study of functions and sequences on $\Bbb R$ and $\Bbb Z$, with no equivalent inequalities in higher-dimensional spaces. The aim of this paper is to obtain a new class of discrete Landau-Kolmogorov type inequalities of arbitrary dimension: $$ \|φ\|_{\ell^\infty(\Bbb Z^d)} \leq μ_{p,d}\|\nabla_Dφ\|^{p/2^d}_{\ell^2(\Bbb Z^d)}\, \|φ\|^{1-p/2^d}_{\ell^2(\Bbb Z^d)}, % $$ where the constant $μ_{p,d}$ is explicitly specified. In fact, this…

Weighted inequalities for discrete iterated Hardy operators

We characterize a three-weight inequality for an iterated discrete Hardy-type operator. In the case when the domain space is a weighted space $\ell^p$ with $p\in(0,1]$, we develop characterizations which enable us to reduce the problem to another one with $p=1$. This, in turn, makes it possible to establish an equivalence of the weighted discrete inequality to an appropriate inequality for iterated Hardy-type operators acting on measurable functions defined on $\mathbb{R}$, for all cases of involved positive exponents.

Weighted inequalities for discrete iterated kernel operators

We develop a new method that enables us to solve the open problem of characterizing discrete inequalities for kernel operators involving suprema. More precisely, we establish necessary and sufficient conditions under which there exists a positive constant $C$ such that \begin{equation*} \Bigg(\sum_{n\in\mathbb{Z}}\Bigg(\sum_{i=-\infty}^n {U}(i,n) a_i\Bigg)^{q} {w}_n\Bigg)^{\frac{1}{q}} \le C \Bigg(\sum_{n\in\mathbb{Z}}a_n^p {v}_n\Bigg)^{\frac{1}{p}} \end{equation*} holds for every sequence of nonnegative numbers $\{a_n\}_{n\in\mathbb{Z}}$ where $U$ is a kernel satisfying certain regularity con…

Generalization of the Ehrling inequality and universal characterization of completely continuous operators

The present work is devoted to an extension of the well-known Ehrling inequalities, which quantitatively characterize compact embeddings of function spaces, to more general operators. Firstly, a modified notion of continuity for linear operators, named \emph{Ehrling continuity} and inspired by the classical Ehrling inequality, is introduced, and then, a necessary and sufficient condition for Ehrling continuity is provided via arguments based on general topology. Secondly, general completely continuous operators between normed spaces are characterized in terms of (generalized) Ehrling type ineq…

三角混沌与$\mathrm{X}_p$不等式 I——离散群上的平衡Fourier截断

Trigonometric chaos and $\mathrm{X}_p$ inequalities I -- Balanced Fourier truncations over discrete groups

研究群代数中Fourier截断平衡平均的$L_p$估计，将Naor关于超立方体的基本不等式推广至离散群，利用非交换$L_p$空间的Banach $\mathrm{X}_p$性质和非交换Riesz变换的无维数估计进行证明，并在自由群情形采用基于自由Hilbert变换的替代方法。

Discrete Weighted Hardy Inequalities with Different Kinds of Boundary Conditions

This paper studies the weighted Hardy inequalities on the discrete intervals with four different kinds of boundary conditions. The main result is the uniform expression of the basic estimate of the optimal constant with the corresponding boundary condition. Firstly, one-side boundary condition is considered, which means that the sequences vanish at the right endpoint (ND-case). Based on the dual method, it can be translated into the case vanishing at left endpoint (DN-case). Secondly, the condition is the case that the sequences vanish at two endpoints (DD-case). The third type of condition is…

New Constants in Discrete Lieb-Thirring Inequalities for Jacobi Matrices

This paper is essentially derived from the observation that some results used for improving constants in the Lieb-Thirring inequalities for Schrodinger operators in L2(-\infty,\infty) can be translated to the discrete Schrodinger op- erators and more generally to Jacobi matrices. Some results were previ- ously proved by D. Hundertmark and B. Simon and the aim of this article is to improve the constants obtained in their article [7].

On the optimal constants in the two-sided Stechkin inequalities

We address the optimal constants in the strong and the weak Stechkin inequalities, both in their discrete and continuous variants. These inequalities appear in the characterization of approximation spaces which arise from sparse approximation or have applications to interpolation theory. An elementary proof of a constant in the strong discrete Stechkin inequality given by Bennett is provided, and we improve the constants given by Levin and Stechkin and by Copson. Finally, the minimal constants in the weak discrete Stechkin inequalities and both continuous Stechkin inequalities are presented.

有偏离散立方体上的向量值集中不等式

Vector-valued concentration inequalities on the biased discrete cube

该研究提出离散超立方体有偏测度下对偏差参数和目标巴拿赫空间Rademacher型具最优依赖的向量值集中不等式，推导出泊松乘积测度的相关新不等式，并得到超立方体嵌入非平凡型巴拿赫空间的平均畸变下界及平均不可嵌入结论。

Uncertainty inequalities on groups and homogeneous spaces via isoperimetric inequalities

We prove a family of $L^p$ uncertainty inequalities on fairly general groups and homogeneous spaces, both in the smooth and in the discrete setting. The crucial point is the proof of the $L^1$ endpoint, which is derived from a general weak isoperimetric inequality.

半直线上的最优分数阶离散Hardy不等式

Optimal fractional discrete Hardy inequalities on the half-line

该研究将半直线上分数阶离散拉普拉斯算子的Toeplitz实现视为全空间算子的压缩，证明了强最优的分数阶Hardy不等式，且等号无法由非零元达到，进而得到离散Birman不等式的分数阶推广。

整数集上的一维离散Hardy与Rellich不等式

One-dimensional discrete Hardy and Rellich inequalities on integers

该研究针对整数集上的一维离散Hardy和Rellich不等式，证明了偶自然数幂权下带最佳常数和余项的加权Hardy不等式，得到多类Rellich不等式的显式常数，还通过解析方法推导出组合恒等式，揭示了组合与函数恒等式的关联。

Curvature and transport inequalities for Markov chains in discrete spaces

We study various transport-information inequalities under three different notions of Ricci curvature in the discrete setting: the curvature-dimension condition of Bakry and Émery, the exponential curvature-dimension condition of Bauer \textit{et al.} and the coarse Ricci curvature of Ollivier. We prove that under a curvature-dimension condition or coarse Ricci curvature condition, an $L_1$ transport-information inequality holds; while under an exponential curvature-dimension condition, some weak-transport information inequalities hold. As an application, we establish a Bonnet-Meyer's theorem u…

The non-compact normed space of norms on a finite-dimensional Banach space

We discuss a new pseudometric on the space of all norms on a finite-dimensional vector space (or free module) $\mathbb{F}^k$, with $\mathbb{F}$ the real, complex, or quaternion numbers. This metric arises from the Lipschitz-equivalence of all norms on $\mathbb{F}^k$, and seems to be unexplored in the literature. We initiate the study of the associated quotient metric space, and show that it is complete, connected, and non-compact. In particular, the new topology is strictly coarser than that of the Banach-Mazur compactum. For example, for each $k \geqslant 2$ the metric subspace $\{ \| \cdot \…

Banach space valued Pisier and Riesz type inequalities on discrete cube

This is an attempt to build Banach space valued theory for certain singular integrals on Hamming cube. Of course all estimates below are dimension independent, and we tried to find ultimate sharp assumptions on the Banach space for a corresponding operators to be bounded. In certain cases we succeeded, although there are still many open questions, some of them are listed in the last Section. Using the approach of \cite{IVHV} and also quantum random variables approach of \cite{ELP} we generalize several theorems of Pisier \cite{P} and Hytönen-Naor \cite{HN}. We also improve the constant in $L^1…

离散热流的若干几何应用

Some geometric applications of the discrete heat flow

本研究利用离散超立方体上的热流方法，得到双Lipschitz嵌入畸变下界的改进估计，还提出度量稳定型不变量并证明其与线性稳定型一致，可用于加权超立方体的不可嵌入分析。

一般加权离散p-Hardy不等式中的尖锐形式和定量稳定性

Sharp forms and quantitative stability for general weighted discrete $p$-Hardy inequalities

本文通过选择权重和指定1<p<∞，恢复Krejčiřík-Štampach的结果，获得p-Hardy不等式的尖锐形式，并推广Gupta的加权不等式，同时证明了离散p-Hardy不等式最小序列趋近于非平凡极小解的定量稳定性结果。

非线性谱隙的一种组合方法

A combinatorial approach to nonlinear spectral gaps

针对非线性谱隙的定量估计问题，提出基于正则图长程扩张性质的新颖组合框架，证明Poincaré常数对余型q的依赖为最优的多项式级别，并推广了随机图低失真嵌入的下界结果。

Discrete Riesz transforms and sharp metric $X_p$ inequalities

$ \renewcommand{\subset}{\subseteq} \newcommand{\N}{\mathbb N} $For $p\in [2,\infty)$ the metric $X_p$ inequality with sharp scaling parameter is proven here to hold true in $L_p$. The geometric consequences of this result include the following sharp statements about embeddings of $L_q$ into $L_p$ when $2< q<p<\infty$: the maximal $θ\in (0,1]$ for which $L_q$ admits a bi-$θ$-Hölder embedding into $L_p$ equals $q/p$, and for $m,n\in \N$ the smallest possible bi-Lipschitz distortion of any embedding into $L_p$ of the grid $\{1,\ldots,m\}^n\subset \ell_q^n$ is bounded above and below by constant …

关于有界上同调中无圈性的一些注记

Some remarks on acyclicity in bounded cohomology

本文证明群同态的核有界无圈当且仅当该同态对所有对偶赋范模诱导有界上同调同构，并推广到道路连通空间间的映射，核心是证明有界无圈群对所有对偶赋范平凡模具有平凡有界上同调。

Poincaré, modified logarithmic Sobolev and isoperimetric inequalities for Markov chains with non-negative Ricci curvature

We study functional inequalities for Markov chains on discrete spaces with entropic Ricci curvature bounded from below. Our main results are that when curvature is non-negative, but not necessarily positive, the spectral gap, the Cheeger isoperimetric constant and the modified logarithmic Sobolev constant of the chain can be bounded from below by a constant that only depends on the diameter of the space, with respect to a suitable metric. These estimates are discrete analogues of classical results of Riemannian geometry obtained by Li and Yau, Buser and Wang.

格图上离散函数不等式

Discrete functional inequalities on lattice graphs

本论文研究格图上的离散Hardy型不等式和重排不等式，证明尖锐常数并分析其渐近行为，建立与连续不等式及等周不等式的联系。

Discrete Schwarz rearrangement in lattice graphs

In this paper, we prove a discrete version of the generalized Riesz inequality on $\mathbb{Z}^d$. As a consequence, we will derive the extended Hardy-Littlewood and Pólya-Szegö inequalities. We will also establish cases of equality in the latter. Our approach is totally novel and self-contained. In particular, we invented a definition for the discrete rearrangement in higher dimensions. Moreover, we show that the definition "suggested" by Pruss does not work. We solve a long-standing open question raised by Alexander Pruss in [Pru98, p494], Duke Math Journal, and discussed with him in several …