Mohamed Amine Ben Amor, Amal Omrani
In this work, we will generalize the moment generating function to Riesz spaces. We will derive some of its properties and use it to prove concentration inequalities on Riesz spaces.
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We prove that supermodularity is a necessary condition for the generalized Hardy- Littlewood and Riesz rearrangement inequalities. We also show the necessity of the monotonicity of the kernels involved in the Riesz{type integral.
Bernstein-Nikolskii inequalities and Riesz interpolation formula are established for eigenfunctions of Laplace operators and polynomials on compact homogeneous manifolds.
We provide a new technique to prove Ornstein's non-inequalities for derivatives with some geometrical dependence on their indexes.
In this article we obtain the non - asymptotical low estimations for bilinear Riesz's functional through the Lebesgue spaces norms by means of building of some examples.
New weighted Riesz-type pointwise inequalities and applications to generalized Sobolev estimates
本文提出新的加权里斯型点wise不等式，并将其应用于广义索伯列夫估计中。
Various topics in stochastic processes have been considered in the abstract setting of Riesz spaces, for example martingales, martingale convergence, ergodic theory, AMARTS, Markov processes and mixingales. Here we continue the relaxation of conditional independence begun in the study of mixingales and study mixing processes. The two mixing coefficients which will be considered are the $α$ (strong) and $φ$ (uniform) mixing coefficients. We conclude with mixing inequalities for these types of processes. In order to facilitate this development, the study of generalized $L^1$ and $L^\infty$ space…
In this paper, we develop a thorough analysis of the boundedness properties of the maximal operator for the Bochner-Riesz means related to the Fourier-Bessel expansions. For this operator, we study weighted and unweighted inequalities in the spaces L^p((0,1),x^{2ν+1}dx). Moreover, weak and restricted weak type inequalities are obtained for the critical values of p. As a consequence, we deduce the almost everywhere pointwise convergence of these means.
In this paper we establish $L^p$-boundedness properties for variation, oscillation and jump operators associated with Riesz transforms and Poisson semigroups related to Laguerre polynomial expansions.
Some Inequalities for Riesz Potential on Homogeneous Variable Exponent Herz-Morrey-Hardy Spaces
本文研究调和分析中Riesz位势在齐次变指数Herz-Morrey-Hardy空间上的有界性，并在一定条件下给出了相应不等式。
The abstraction of the study of stochastic processes to Banach lattices and vector lattices has received much attention by Grobler, Kuo, Labuschagne, Stoica, Troitsky and Watson over the past fifteen years. By contrast mixing processes have received very little attention. In particular mixingales were generalized to the Riesz space setting in {\sc W.-C. Kuo, J.J. Vardy, B.A. Watson,} Mixingales on Riesz spaces, {\em J. Math. Anal. Appl.}, \textbf{402} (2013), 731-738. The concepts of strong and uniform mixing as well as related mixing inequalities were extended to this setting in {\sc W.-C. Ku…
The purpose of this paper is to develop some methods to study Riesz type inequalities, Hardy-Littlewood type theorems and smooth moduli of holomorphic, pluriharmonic and harmonic functions in high-dimensional cases. Initially, we prove some sharp Riesz type inequalities of pluriharmonic functions on bounded symmetric domains. The obtained results extend the main results in (\textit{Trans. Amer. Math. Soc.} {\bf 372} (2019)~ 4031--4051). Furthermore, some Hardy-Littlewood type theorems of holomorphic and pluriharmonic functions on John domains are established. Additionally, we also discuss the …
Reverse inequalities for super-Riesz transforms on graphs with a slow diffusion
在D维Vicsek图上，证明了Riesz型不等式$\|\nabla f\|_p \leq C \|\Delta^\gamma f\|_p$对$p\in(1,\infty)$和$0<\gamma<\gamma^*(p)$成立，而对$\gamma>\gamma^*(p)$不成立，临界指数$\gamma^*(p)$处仍开放，首次给出$L^p$有界超Riesz变换的例子。
In this paper we study weighted mixed norm estimates for Riesz transforms associated to Dunkl harmonic oscillators. The idea is to show that the required inequalities are equivalent to certain vector valued inequalities for operator defined in terms of Laguerre expansions. In certain cases the main result can be deduced from the corresponding result for Hermite Riesz transforms.
Decomposition of an $ L^{1}(T) $-bounded martingale and Applications in Riesz spaces
本文提出了一种将 $ L^{1}(T) $-有界鞅分解为三个鞅的方法，并展示了其在向量格随机分析新理论中处理特定不等式问题的应用。
In this note we prove that the ball is a maximiser of some Schatten $p$-norms of the Riesz potential operators among all domains of a given measure in $\mathbb R^{d}$. In particular, the result is valid for the polyharmonic Newton potential operator, which is related to a nonlocal boundary value problem for the poly-Laplacian extending the one considered by M. Kac in the case of the Laplacian, so we obtain and isoperimetric inequalities for its eigenvalues as well, namely, analogues of Rayleigh-Faber-Krahn and Hong-Krahn-Szegö inequalities.
Riesz transform and its related inequalities for degenerate elliptic operators of Grushin type
研究Grushin型退化椭圆算子的Riesz变换\(L^p\)有界性及反向Riesz不等式，通过显式构造泊松核、格林核及调和湮灭方法，证明其在不同退化情形下的有界性，阐明相关现象机制并得出精确不等式。
We study the boundedness of Riesz transforms in $L^p$ for $p>2$ on a doubling metric measure space endowed with a gradient operator and an injective, $ω$-accretive operator $L$ satisfying Davies-Gaffney estimates. If $L$ is non-negative self-adjoint, we show that under a reverse Hölder inequality, the Riesz transform is always bounded on $L^p$ for $p$ in some interval $[2,2+\varepsilon)$, and that $L^p$ gradient estimates for the semigroup imply boundedness of the Riesz transform in $L^q$ for $q \in [2,p)$. This improves results of \cite{ACDH} and \cite{AC}, where the stronger assumption of a …
Trace Principle for Riesz Potentials on Herz-Type Spaces and Applications
该研究建立了Herz型空间上Riesz位势的迹不等式，论证了参数条件的最优性，给出了相关Sobolev型不等式应用，并得到Herz型Sobolev空间的嵌入定理。
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…
Finite Riesz products and Ornstein non-inequalities on quantum tori
本文研究了量子环上有限里斯茨积的构造，并基于此推导出非交换奥本海姆非等式。
Riesz potential estimates under co-canceling constraints
本文在反相抵消约束下，推导了适用于一般排列不变空间族的Riesz势不等式，揭示了该性质不仅限于L¹空间。
Sobolev type inequalities for fractional maximal functions and Riesz potentials in half spaces
本文研究了半空间中双相泛函加权 Morrey 空间内分数次极大函数与 Riesz 势的 Sobolev 型不等式，并证明了 Riesz 势算子及 Sobolev 函数向加权 Campanato 空间的嵌入性质。
In this paper we obtain the non - asymptotic estimations for Riesz's and Bessel's potential integral operators in the so - called Bilateral Grand Lebesgue Spaces. We also give examples to show the sharpness of these inequalities.
The rearrangement inequalities of Hardy-Littlewood and Riesz say that certain integrals involving products of two or three functions increase under symmetric decreasing rearrangement. It is known that these inequalities extend to integrands of the form F(u_1,..., u_m) where F is supermodular; in particular, they hold when F has nonnegative mixed second derivatives. This paper concerns the regularity assumptions on F and the equality cases. It is shown here that extended Hardy-Littlewood and Riesz inequalities are valid for supermodular integrands that are just Borel measurable. Under some nond…
Let (E,H,mu) be an abstract Wiener space and let D_V := VD, where D denotes the Malliavin derivative and V is a closed and densely defined operator from H into another Hilbert space G. Given a bounded operator B on G, coercive on the closure of the range of V, we consider the realisation of the operator D_V* B D_V in L^p(E,mu) for 1<p<\infty. Our main result states that the following assertions are equivalent: (1) dom((sqrt(D_V* B D_V)) = dom(D_V) and Meyer's inequalities hold for D_V* B D_V; (2) D_V D_V* B admits a bounded H-infinity calculus on the closure of the range of D_V; (3) dom(sqrt(V…
On the Riesz transform and its reverse inequality on manifolds with quadratically decaying curvature
本文研究了曲率二次衰减流形上的Riesz变换及其反向不等式，通过Lorentz型估计和谐波环消方法，证明了反向Riesz、Hardy和加权Sobolev不等式等价，并验证了Grushin空间中Riesz变换的有界性。
$ \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 …
The aim of this paper is to prove an improved version of the bounded differences inequality for matrix valued functions, by developing the methods of Mackey et al.: "Matrix Concentration Inequalities via the Method of Exchangeable Pairs". Along the way, we prove new trace inequalities for the matrix exponential.
In this paper we study the existence of maximizers for two families of interpolation inequalities, namely a generalized Gagliardo-Nirenberg inequality and a new inequality involving the Riesz energy. Two basic tools in our argument are a generalization of Lieb's Translation Lemma and a Riesz energy version of the Brézis--Lieb lemma.
Localized frames without inequalities
本文证明可分希尔伯特空间中相互局部化的可数向量族，其框架性质等价于九个不涉及不等式的条件，并应用于平移不变空间得到稳定采样集的新条件。
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。
The martingale method is used to establish concentration inequalities for a class of dependent random sequences on a countable state space, with the constants in the inequalities expressed in terms of certain mixing coefficients. Along the way, bounds are obtained on martingale differences associated with the random sequences, which may be of independent interest. As applications of the main result, concentration inequalities are also derived for inhomogeneous Markov chains and hidden Markov chains, and an extremal property associated with their martingale difference bounds is established. Thi…
We are interested in Sobolev type inequalities and their relationship with concentration properties on higher dimensions. We consider unbounded spin systems on the d-dimensional lattice with interactions that increase slower than a quadratic. At first we assume that the one site measure satisfies a Modified log-Sobolev inequality with a constant uniformly on the boundary conditions and we determine conditions so that the infinite dimensional Gibbs measure satisfies a concentration as well as a Talagrand type inequality. Then a Modified Log-Sobolev type concentration property is obtained under …
For a real-valued measurable function $f$ and a nonnegative, nondecreasing function $ϕ$, we first obtain a Chebyshev type inequality which provides an upper bound for $\displaystyle ϕ(λ_{1}) μ(\{x \in Ω: f(x) \geq λ_{1} \}) + \sum_{k=2}^{n}\left(ϕ(λ_{k})- ϕ(λ_{k-1})\right) μ(\{x \in Ω: f(x) \geq λ_{k}\}) ,$ where $0 < λ_1 < λ_2 \cdots λ_n < \infty$. Using this, generalizations of a few concentration inequalities such as Markov, reverse Markov, Bienaymé-Chebyshev, Cantelli and Hoeffding inequalities are obtained.
We prove a sharp inequality conjectured by Bobkov on the measure of dilations of Borel sets in $\mathbb{R}^n$ by a $s$-concave probability. Our result gives a common generalization of an inequality of Nazarov, Sodin and Volberg and a concentration inequality of Guédon. Applying our inequality to the level sets of functions satisfying a Remez type inequality, we deduce, as it is classical, that these functions enjoy dimension free distribution inequalities and Kahane-Khintchine type inequalities with positive and negative exponent, with respect to an arbitrary $s$-concave probability.
Best constants in inequalities involving analytic and co-analytic projections and Riesz theorem for various function spaces
本文研究了用Lebesgue $p$-范数估计Riesz投影算子表达式的精确不等式，改进了已有结果并证明了相关猜想，同时导出了多种函数空间上的Riesz型定理。
Sharp pointwise gradient estimates for Riesz potentials with a bounded density
本文在$\mathbb{R}^{n}$中建立了具有有界密度的Riesz位势及其梯度的锐点态不等式，并说明其在位势分析、矩理论等领域的应用价值。
We analyse Morrey spaces, generalised Morrey spaces and Campanato spaces on homogeneous groups. The boundedness of the Hardy-Littlewood maximal operator, Bessel-Riesz operators, generalised Bessel-Riesz operators and generalised fractional integral operators in generalised Morrey spaces on homogeneous groups is shown. Moreover, we prove the boundedness of the modified version of the generalised fractional integral operator and Olsen type inequalities in Campanato spaces and generalised Morrey spaces on homogeneous groups, respectively. Our results extend results known in the isotropic Euclidea…
We define Hölder classes $Λ_α$ associated with a Markovian semigroup and prove that, when the semigroup satisfies the $Γ^2 \geq 0$ condition, the Riesz transforms are bounded between the Hölder classes. As a consequence, this bound holds in manifolds with nonnegative Ricci curvature. We also show, without the need for extra assumptions on the semigroup, that a version of the Morrey inequalities is equivalent to the ultracontractivity property. This result extends the semigroup approach to the Sobolev inequalities laid by Varopoulos. After that, we study certain families of operators between th…
The main aim of this paper is to investigate $\left( H_{p},L_{p}\right) $ and $\left( H_{p},L_{p,\infty }\right) $ type inequalities for maximal operators of Riesz logarithmic means of one-dimensional Vilenkin-Fourier series.
On Gilles Pisier's approach to gaussian concentration, isoperimetry and Poincaré-type inequalities
本研究拓展了Gilles Pisier的研究方法，可在欧氏空间旋转不变概率分布（含多维柯西测度）中探索高斯测度相关集中、等周与庞加莱型不等式结果的对应形式。
We establish a good lambda inequality relating to the distribution function of Riesz potential and fractional maximal function on $\left(\mathbb{R}^n, dμ\right)$ where $μ$ is a positive Radon measure which doesn't necessarily satisfy a doubling condition. This is extended to weights $w$ in $A_{\infty}(μ)$ associated to the measure $μ$. We also derive potential inequalities as an application.
We consider generalized Orlicz-Morrey spaces $M_{Φ,φ}(\Rn)$ including their weak versions $WM_{Φ,φ}(\Rn)$. In these spaces we prove the boundedness of the Riesz potential from $M_{Φ,φ_1}(\Rn)$ to $M_{Ψ,φ_2}(\Rn)$ and from $M_{Φ,φ_1}(\Rn)$ to $WM_{Ψ,φ_2}(\Rn)$. As applications of those results, the boundedness of the commutators of the Riesz potential on generalized Orlicz-Morrey space is also obtained. In all the cases the conditions for the boundedness are given either in terms of Zygmund-type integral inequalities on $(φ_{1},φ_{2})$, which do not assume any assumption on monotonicity of $φ_{…
We characterize the dual spaces of the generalized Hardy spaces defined by replacing Lebesgue quasi-norms by Wiener amalgam ones. In these generalized Hardy spaces, we prove that some classical linear operators such as Calderón-Zygmund, convolution and Riesz potential operators are bounded.
Growth estimates on optimal transport maps via concentration inequalities
本文通过集中不等式和映射单调性，为Brenier映射的多项式上界提供替代证明并扩展结果，涵盖从高斯到对数凹测度的二次增长情形。
Contractive inequalities between Dirichlet and Hardy spaces
本文证明了 Brevig 等人关于 Dirichlet 空间与 Hardy 空间之间压缩不等式的猜想，并探讨了其与 Riesz 投影的联系。
We state and discuss several interrelated results, conjectures, and questions regarding contractive inequalities for classical $H^p$ spaces of the unit disc. We study both coefficient estimates in terms of weighted $\ell^2$ sums and the Riesz projection viewed as a map from $L^q$ to $H^p$ with $q\ge p$. Some numerical evidence is given that supports our conjectures.
We generalize several theorems of Hytönen-Naor \cite{HN} using the approach from \cite{IVHV}. In particular, we give yet another necessary and sufficient condition (see (3.2)) to be a $K$-convex space, where the sufficiency was proved by Naor--Schechtman \cite{NS}. This condition is in terms of the boundedness of the second order Riesz transforms $\{Δ^{-1} D_i\}_{i=1}^n$ in $L^p(Ω_n, X)$.
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