Sharp Hardy type inequalities on the Carnot Group

Existence of optimizers of the Stein-Weiss inequalities on Carnot groups

This paper proves existence of optimizers of the Stein-Weiss inequalities on Carnot groups under some conditions. The adjustment of Lions' concentration compactness principles to Carnot groups plays an important role in our proof. Unlike known treatment to the Hardy-Littlewood-Sobolev inequality on Heisenberg group, our arguments relate to the powers of the weight functions.

Hardy and Hardy PDO type inequalities in domains. Part I

The previous "Polynomial Capacities, Poincare' type inequalities and Spectral synthesis in Sobolev space" is a prerequisite. A parallell reading is recommended.

A remark on the history of Hardy inequalities in domains

This a very brief account of the main line of development of Hardy inequalities.

Two limits on Hardy and Sobolev inequalities

It is known that classical Hardy and Sobolev inequalities hold when the exponent $p$ and the dimension $N$ satisfy $p < N < \infty$. In this note, we consider two limits of Hardy and Sobolev inequalities as $p \nearrow N$ and $N \nearrow \infty$ in some sense.

Improved Hardy and Rellich inequalities on Riemannian manifolds

In this paper we establish improved Hardy and Rellich type inequalities on Riemannian manifold $M$. Furthermore, we also obtain sharp constant for the improved Hardy inequality and explicit constant for the Rellich inequality on hyperbolic space $\mathbb{H}^n$.

Triebel-Lizorkin空间中Hardy不等式最佳常数

Best constants for Hardy inequalities in Triebel--Lizorkin spaces

本文研究了Triebel-Lizorkin空间中分数Hardy不等式最佳常数的问题，推广了加权分数Hardy不等式并给出了新的结果。

卡诺群上的强制不等式：驯服奇异性

Coercive Inequalities on Carnot Groups: Taming Singularities

本文在卡诺群上提出通过引入自然奇异性来获得强制不等式的方法，构造了保证庞加莱和对数索博列夫不等式的概率测度，并应用于海森堡群上的 Dyson-Ornstein-Uhlenbeck 模型及谱离散性研究。

Poincaré and Hardy inequalities on homogeneous trees

We study Hardy-type inequalities on infinite homogeneous trees. More precisely, we derive optimal Hardy weights for the combinatorial Laplacian in this setting and we obtain, as a consequence, optimal improvements for the Poincaré inequality.

Hardy-Littlewood-Sobolev and related inequalities: stability

The purpose of this text is twofold. We present a review of the existing stability results for Sobolev, Hardy-Littlewood-Sobolev (HLS) and related inequalities. We also contribute to the topic with some observations on constructive stability estimates for (HLS).

Weighted $L^p$-Hardy and $L^p$-Rellich inequalities with boundary terms on stratified Lie groups

In this paper, generalised weighted $L^p$-Hardy,$ L^p$-Caffarelli-Kohn-Nirenberg, and $L^p$-Rellich inequalities with boundary terms are obtained on stratified Lie groups. As consequences, most of the Hardy type inequalities and Heisenberg- Pauli-Weyl type uncertainty principles on stratified groups are recovered. Moreover, a weighted $L^2$-Rellich type inequality with the boundary term is obtained.

Sharp subcritical and critical $L^{p}$ Hardy inequalities on the sphere

We prove sharp inequalities of Hardy type for functions in the Sobolev space $W^{1,p}$ on the unit sphere $\mathbb{S}^{n-1}$ in $\mathbb{R}^{n}$. We achieve this in both the subcritical and critical cases. The method we use to show optimality takes into account all the constants involved in our inequalities.

Sharp Hardy-Littlewood-Sobolev Inequalities on Octonionic Heisenberg Group

This paper is a second one following our work [CLZ13] in series, considering sharp Hardy- Littlewood-Sobolev inequalities on groups of Heisenberg type. The first important breakthrough was made by Frank and Lieb in [FL12]. In this paper, analogous results are obtained for octonionic Heisenberg group.

Sharp fractional Hardy inequalities in half-spaces

We determine the sharp constant in the Hardy inequality for fractional Sobolev spaces on half-spaces. Our proof relies on a non-linear and non-local version of the ground state representation.

A survey of Hardy type inequalities on homogeneous groups

In this review paper, we survey Hardy type inequalities from the point of view of Folland and Stein's homogeneous groups. Particular attention is paid to Hardy type inequalities on stratified groups which give a special class of homogeneous groups. In this environment, the theory of Hardy type inequalities becomes intricately intertwined with the properties of sub-Laplacians and more general subelliptic partial differential equations. Particularly, we discuss the Badiale-Tarantello conjecture and a conjecture on the geometric Hardy inequality in a half-space of the Heisenberg group with a shar…

Hardy不等式与非局部容量

Hardy inequalities and nonlocal capacity

本文引入并研究了与非局部Sobolev空间相关的容量，特别是零阶非局部算子对应的空间，通过证明Hardy型不等式获得Sobolev嵌入，进而估计球体的非局部容量。

反对称函数上的Hardy不等式与Sobolev不等式

Hardy and Sobolev inequalities on antisymmetric functions

该研究将一维粒子反对称函数的Hardy不等式拓展至多维粒子情形，得到了精确的Hardy不等式，还附带得到若干Sobolev型与Gagliardo-Nirenberg型不等式，可用于研究Schrödinger算子的谱性质。

Subelliptic geometric Hardy type inequalities on half-spaces and convex domains

In this paper we present $L^2$ and $L^p$ versions of the geometric Hardy inequalities in half-spaces and convex domains on stratified (Lie) groups. As a consequence, we obtain the geometric uncertainty principles. We give examples of the obtained results for the Heisenberg and the Engel groups.

On the best constant of Hardy-Sobolev Inequalities

We obtain the sharp constant for the Hardy-Sobolev inequality involving the distance to the origin. This inequality is equivalent to a limiting Caffarelli-Kohn-Nirenberg inequality. In three dimensions, in certain cases the sharp constant coincides with the best Sobolev constant.

One dimensional Weighted Hardy's Inequalities and application

In the present paper we shall improve one dimensional weighted Hardy inequalities with one-sided boundary condition by adding sharp remainders. As an application, we shall establish n dimensional weighted Hardy inequalities in a bounded smooth domain with weight functions being powers of the distance function d(x) to the boundary. Our results will be applicable to variational problems in a coming paper.

On horizontal Hardy, Rellich, Caffarelli-Kohn-Nirenberg and $p$-sub-Laplacian inequalities on stratified groups

In this paper, we present a version of horizontal weighted Hardy-Rellich type and Caffarelli-Kohn-Nirenberg type inequalities on stratified groups and study some of their consequences. Our results reflect on many results previously known in special cases. Moreover, a new simple proof of the Badiale-Tarantello conjecture [2] on the best constant of a Hardy type inequality is provided. We also show a family of Poincaré inequalities as well as inequalities involving the weighted and unweighted $p$-sub-Laplacians.

$q$-Poincaré 不等式在具有 filiform 型李代数的 Carnot 群上

$q$-Poincaré inequalities on Carnot Groups with filiform type Lie algebra

本文证明了在 filiform 型李代数的幂零李群上，概率测度的全局 $q$-Poincaré 不等式，其中测度密度依赖于齐次范数。

Carnot群上的Poincaré不等式及Schrödinger算子的谱间隙

Poincaré inequalities on Carnot Groups and spectral gap of Schrödinger operators

本文研究了Carnot群上概率测度下的Poincaré不等式条件，并证明了相关Schrödinger算子具有谱间隙。

Spectral Analysis for Perturbed Operators on Carnot Groups

Let $\G$ be a Carnot group of homogeneous dimension $M$ and $Δ$ its horizontal sublaplacian. For $α\in(0,M)$ we show that operators of the form $H_α:=(-Δ)^α+V$ have no singular spectrum, under generous assumptions on the multiplication operator $V$. The proof is based on commutator methods and Hardy inequalities.

(Weak) Hardy and Poincaré inequalities and criticality theory

In this paper we study Hardy and Poincaré inequalities and their weak versions for quadratic forms satisfying the first Beurling-Deny criterion. We employ these inequalities to establish a criticality theory for such forms.

Some trace Hardy type inequalities and trace Hardy-Sobolev-Maz'ya type inequalities

We prove a trace Hardy type inequality with the best constant on the polyhedral convex cones which generalizes recent results of Alvino et al. and of Tzirakis on the upper half space. We also prove some trace Hardy-Sobolev-Maz'ya type inequalities which generalize the recent results of Filippas et al.. In applications, we derive some Hardy type inequalities and Hardy-Sobolev-Maz'ya type inequalities for fractional Laplacian. Finally, we prove the logarithmic Sobolev trace inequalities and logarithmic Hardy trace inequalities on the upper half spaces. The best constants in these inequalities ar…

Hardy-type inequalities for Dunkl operators with applications to many-particle Hardy inequalities

In this paper we study various forms of the Hardy inequality for Dunkl operators, including the classical inequality, $L^p$ inequalities, an improved Hardy inequality, as well as the Rellich inequality and a special case of the Caffarelli-Kohn-Nirenberg inequality. As a consequence, one-dimensional many-particle Hardy inequalities for generalised root systems are proved, which in the particular case of root systems $A_{N-1}$ improve some well-known results.

反对称函数与奇函数的改进Hardy和Rellich不等式

Improved Hardy and Rellich inequalities for antisymmetric and odd functions

本文研究了一般指数p下反对称与奇函数类的Hardy和Rellich不等式，得出了优于经典常数的改进结果。

Hardy inequalities on metric measure spaces

In this note we give several characterisations of weights for two-weight Hardy inequalities to hold on general metric measure spaces possessing polar decompositions. Since there may be no differentiable structure on such spaces, the inequalities are given in the integral form in the spirit of Hardy's original inequality. We give examples obtaining new weighted Hardy inequalities on $\mathbb R^n$, on homogeneous groups, on hyperbolic spaces, and on Cartan-Hadamard manifolds.

Hardy and Rellich inequalities, identities, and sharp remainders on homogeneous groups

We give sharp remainder terms of $L^{p}$ and weighted Hardy and Rellich inequalities on one of most general subclasses of nilpotent Lie groups, namely the class of homogeneous groups. As consequences, we obtain analogues of the generalised classical Hardy and Rellich inequalities and the uncertainty principle on homogeneous groups. We also prove higher order inequalities of Hardy-Rellich type, all with sharp constants. A number of identities are derived including weighted and higher order types.

Sobolev-Bregman形式的精确Hardy不等式

Sharp Hardy inequalities for Sobolev-Bregman forms

本文研究了Sobolev-Bregman形式中半空间和凸域的精确分数Hardy不等式，扩展了Bogdan和Dyda以及Loss和Sloane的结果。

Hardy-Sobolev-Maz'ya inequalities for arbitrary domains

We prove a Hardy-Sobolev-Maz'ya inequality for arbitrary domains Ω\subset\R^N with a constant depending only on the dimension N\geq 3. In particular, for convex domains this settles a conjecture by Filippas, Maz'ya and Tertikas. As an application we derive Hardy-Lieb-Thirring inequalities for eigenvalues of Schrödinger operators on domains.

Necessary Conditions for Fractional Hardy-Sobolev's Inequalities

In this short article we obtain some necessary conditions for a so-called fractional Hardy-Sobolev's inequalities in multidimensional case. We also give some examples to show the sharpness of these inequalities.

On weighted remainder form of Hardy-type inequalities

We use different approaches to study a generalization of a result of Levin and Stečkin concerning an inequality analogous to Hardy's inequality. Our results lead naturally to the study of weighted remainder form of Hardy-type inequalities.

广义仿射Hardy-Littlewood-Sobolev不等式

Generalized affine Hardy-Littlewood-Sobolev inequalities

该研究建立了涉及两个不同函数、强于经典形式的仿射Hardy-Littlewood-Sobolev不等式，并证明了对数凹函数下该新不等式的反向不等式。

Hardy-Poincaré, Rellich and Uncertainty principle inequalities on Riemannian manifolds

We continue our previous study of improved Hardy, Rellich and Uncertainty principle inequalities on a Riemannian manifold $M$, started in \cite{Kombe-Ozaydin}. In the present paper we prove new weighted Hardy-Poincaré, Rellich type inequalities as well as improved version of our Uncertainty principle inequalities on a Riemannian manifold $M$. In particular, we obtain sharp constants for these inequalities on the hyperbolic space $\mathbb{H}^n$.

Fractional Hardy-type inequalities in domains with plump complement

We establish fractional Hardy-type inequalities in a bounded domain with plump complement. In particular our results apply in bounded C^\infty domains and Lipschitz domains.

Limiting Sobolev and Hardy inequalities on stratified homogeneous groups

We give a sufficient condition for limiting Sobolev and Hardy inequalities to hold on stratified homogeneous groups. In the Euclidean case, this condition reduces to the known cancelling necessary and sufficient condition. We obtain in particular endpoint Korn--Sobolev and Korn--Hardy inequalities on stratified homogeneous groups.

Sharp Hardy-Rellich Type Inequalities Associated with Dunkl Operators

In this paper, we obtained the Dunkl analogy of classical Lp Hardy inequality for $p > N + 2γ$ with sharp constant $\left(\frac{p-N-2γ}{p}\right)^{p}$, where $2γ$ is the degree of weight function associated with Dunkl operators, and $L^p$ Hardy inequalities with distant function in some G-invariant domains. Moreover we proved two sharp Hardy-Rellich type inequalities for Dunkl operators.

Hardy inequalities on constant-order noncommutative Vilenkin groups

In this note we extend several integral inequalities to the context of noncommutative Vilenkin groups. We prove some sharp weak and strong type estimates for the Hardy operator and the Hardy-Littlewood-P{ó}lya operator on constant-order noncommutative Vilenkin groups. In particular for graded $\K$-Lie groups, where $\K$ is a non-archimedean local field, we additionally provide some functional inequalities, like the Hardy-Littlewood-Sobolev unequality and the Stein-Weiss inequality, linking some classes of homogeneous pseudo-differential operators, like the Vladimirov-Taibleson operator and the…

On the necessity of the assumptions used to prove Hardy-Littlewood and Riesz Rearrangement Inequalities 1

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.

From Hardy to Rellich inequalities on graphs

We show how to deduce Rellich inequalities from Hardy inequalities on infinite graphs. Specifically, the obtained Rellich inequality gives an upper bound on a function by the Laplacian of the function in terms of weighted norms. These weights involve the Hardy weight and a function which satisfies an eikonal inequality. The results are proven first for Laplacians and are extended to Schrödinger operators afterwards.

Factorizations and Hardy-Rellich inequalities on stratified groups

In this paper, we obtain Hardy, Hardy-Rellich and refined Hardy inequalities on general stratified groups and weighted Hardy inequalities on general homogeneous groups using the factorization method of differential operators, inspired by the recent work of Gesztesy and Littlejohn \cite{GL17}. We note that some of the obtained inequalities are new also in the usual Euclidean setting. We also obtain analogues of Gesztesy and Littlejohn's 2-parameter version of the Rellich inequality on stratified groups and on the Heisenberg group, and a new two-parameter estimate on $\mathbb{R}^{n}$ which can b…

Abstract framework for John Nirenberg inequalities and applications to Hardy spaces

In this paper, we develop an abstract framework for John-Nirenberg inequalities associated to BMO-type spaces. This work can be seen as the sequel of [5], where the authors introduced a very general framework for atomic and molecular Hardy spaces. Moreover, we show that our assumptions allow us to recover some already known John-Nirenberg inequalities. We give applications to the atomic Hardy spaces too.

A weighted anisotropic Sobolev type inequality and its applications to Hardy inequalities

In this paper we focus our attention on an embedding result for a weighted Sobolev space that involves as weight the distance function from the boundary taken with respect to a general smooth gauge function $F$. Starting from this type of inequalities we prove some refined Hardy-type inequalities.

统一的加权Hardy型不等式

Unified weighted Hardy-type inequalities

本文提出一种统一方法，在幂零Lie群上获得具有最佳常数的Hardy型不等式，改进了经典Euclidean设定下的已知结果，并推导出新的Caffarelli-Kohn-Nirenberg型不等式。

一般李群上的Hardy-Sobolev-Rellich、Hardy-Littlewood-Sobolev与Caffarelli-Kohn-Nirenberg不等式

Hardy-Sobolev-Rellich, Hardy-Littlewood-Sobolev and Caffarelli-Kohn-Nirenberg inequalities on general Lie groups

该研究在一般连通李群的次椭圆微分算子框架下，建立了多类几何不等式及其临界形式，覆盖不同拓扑与模性情形，并推导了对应的不确定型原理。

Euler semigroup, Hardy-Sobolev and Gagliardo-Nirenberg type inequalities on homogeneous groups

In this paper we describe the Euler semigroup $\{e^{-t\mathbb{E}^{*}\mathbb{E}}\}_{t>0}$ on homogeneous Lie groups, which allows us to obtain various types of the Hardy-Sobolev and Gagliardo-Nirenberg type inequalities for the Euler operator $\mathbb{E}$. Moreover, the sharp remainder terms of the Sobolev type inequality, maximal Hardy inequality and $|\cdot|$-radial weighted Hardy-Sobolev type inequality are established.

Sobolev, Hardy, Gagliardo-Nirenberg and Caffarelli-Kohn-Nirenberg type inequalities for some fractional derivatives

In this paper we show different inequalities for fractional order differential operators. In particular, the Sobolev, Hardy, Gagliardo-Nirenberg and Caffarelli-Kohn-Nirenberg type inequalities for the Caputo, Riemann-Liouville and Hadamard derivatives are obtained. In addition, we show some applications of these inequalities.

Two-Weighted Inequalities for Hardy-Littlewood Maximal Functions and Singular Integrals in $l^{p(\cdot)}$ Spaces

Two-weight criteria of various type for the Hardy-Littlewood maximal operator and singular integrals in variable exponent Lebesgue spaces defined on the real line are established.

New equivalence theorems for weighted inequalities involving the composition of monotone quasilinear operators with the Hardy and Copson operators and their applications

In this paper, new equivalence theorems for the boundedness of the composition of a quasilinear operator $T$ with the Hardy and Copson operators in weighted Lebesgue spaces are proved. The usefulness of the obtained results is illustrated in the case of weighted Hardy-type and weighted iterated Hardy-type inequalities.

Iterated Hardy-type inequalities involving suprema

In this paper the complete solution of the restricted inequalities for supremal operators are given. The boundedness of the composition of supremal operators with the Hardy and Copson operators in weighted Lebesgue spaces are characterized.

Hardy inequalities with boundary singularities

In this work we prove some Hardy-Poincaré inequalities with quadratic singular potentials localized on the boundary of a smooth domain. Then, we consider conical domains with vertex on the singularity and we show upper and lower bounds for the corresponding optimal constants in the Hardy inequality. In particular, we prove the asymptotic behavior of the optimal constant when the slot of the cone tends to zero.

The sharp higher order Hardy--Rellich type inequalities on the homogeneous groups

We prove several interesting equalities for the integrals of higher order derivatives on the homogeneous groups. As consequences, we obtain the sharp Hardy--Rellich type inequalities for higher order derivatives including both the subcritical and critical inequalities on the homogeneous groups. We also prove several uncertainty principles on the homogeneous groups. Our results seem to be new even in the case of Euclidean space $\mathbb R^n$ and give a simple proof of several classical Hardy--Rellich type inequalities in $\mathbb R^n$.

Hardy-Littlewood inequalities for multipolynomials

The notion of multipolynomials was recently introduced and explored by T. Velanga in [10] as an attempt to encompass the theories of polynomials and multi- linear operators. In the present paper we push this subject further, by proving Hardy- Littlewood inequalities for multipolynomials and, en passant, a variant of the Kahane- Salem-Zygmund inequality in this framework.