The best constant in a fractional Hardy inequality

Multilinear Embedding and Hardy's Inequality

Multilinear trace restriction inequalities are obtained for Hardy's inequality. More generally, detailed development is given for new multilinear forms for Young's convolution inequality, and a new proof for the multilinear Hardy-Littlewood-Sobolev inequality, and its realization on hyperbolic space.

Reverse integral Hardy inequality on metric measure spaces

In this note, we obtain a reverse version of the integral Hardy inequality on metric measure spaces. Moreover, we give necessary and sufficient conditions for the weighted reverse Hardy inequality to be true. The main tool in our proof is a continuous version of the reverse Minkowski inequality. Also, we present some consequences of the obtained reverse Hardy inequality on the homogeneous groups, hyperbolic spaces and Cartan-Hadamard manifolds.

Hardy's Inequality from a Logarithmic Caccioppoli Estimate

We give a simple proof of Hardy's inequality, based on the logarithmic Caccioppoli estimate for p-superharmonic functions in several variables.

A note on higher order fractional Hardy-Sobolev inequalities

We establish some qualitative properties of minimizers in the fractional Hardy--Sobolev inequalities of arbitrary order.

带权重的分数Hardy-Sobolev和Hardy-Sobolev-Maz'ya不等式及其奇点在平坦子流形上的应用

Weighted fractional Hardy-Sobolev and Hardy-Sobolev-Maz'ya inequalities with singularities on flat submanifold

本文研究了带权重的分数Hardy-Sobolev和Hardy-Sobolev-Maz'ya不等式，探讨了奇点在平坦子流形上的应用，并证明了相关不等式在特定条件下的不成立性。

仿射联络的若干Hardy型与Rellich型不等式

Some Hardy and Rellich type inequalities for affine connections

该文研究完备非紧黎曼流形上仿射联络的多种Hardy型及相关不等式，所得结果改进并包含了诸多已有结果作为特例。

Hardy inequalities in fractional Orlicz-Sobolev spaces

In this article we prove both norm and modular Hardy inequalities for a class functions in one-dimensional fractional Orlicz-Sobolev spaces.

Hardy's Inequality for the fractional powers of Grushin operator

We prove Hardy's inequality for the fractional powers of the generalized sublaplacian and the fractional powers of the Grushin operator. We also find an integral representation and a ground state representation for the fractional powers of generalized sublaplacian.

Semilinear elliptic equations for the fractional Laplacian with Hardy potential

In this paper we study existence and nonexistence of nonnegative distributional solutions for a class of semilinear fractional elliptic equations involving the Hardy potential.

精确的分数Hardy不等式及其余项对于1<p<2

Sharp fractional Hardy inequalities with a remainder for $1<p<2$

本文研究了1<p<2范围内带有余项的分数Hardy不等式及其扩展形式

The role of the mean curvature in a Hardy-Sobolev trace inequality

The Hardy-Sobolev trace inequality can be obtained via Harmonic extensions on the half-space of the Stein and Weiss weighted Hardy-Littlewood-Sobolev inequality. In this paper we consider a bounded domain and study the influence of the boundary mean curvature in the Hardy-Sobolev trace inequality on the underlying domain. We prove existence of minimizers when the mean curvature is negative at the singular point of the Hardy potential.

Fractional Laplacian with Hardy potential

We give sharp two-sided estimates of the semigroup generated by the fractional Laplacian plus the Hardy potential on $\mathbb{R}^d$, including the case of the critical constant. We use Davies' method back-to-back with a new method of integral analysis of Duhamel's formula.

Remarks on a Sobolev-Hardy inequality

We compute the optimal constant for a generalized Hardy-Sobolev inequality, and using the product of two symmetrizations we present an elementary proof of the symmetries of some optimal functions. This inequality was motivated by a nonlinear elliptic equation arising in astrophysics.

An optimal Hardy-Morrey inequality

In this work we improve the sharp Hardy inequality in the case $p>n$ by adding an optimal weighted Hoelder semi-norm. To achieve this we first obtain a local improvement. We also obtain a refinement of both the Sobolev inequality for $p>n$ and the Hardy inequality, the latter having the best constant.

On existence of minimizers for the Hardy-Sobolev-Maz'ya inequality

We show existence of minimizers for the Hardy-Sobolev-Maz'ya inequality in $R^{m+n}\setminus\R^n$ for $m=1$ and $n>2$ or for $m>2$ and $n>0$.

Hardy-type and Heisenberg's inequality in Morrey spaces

We use the Morrey norm estimate for the imaginary power of the Laplacian to prove an interpolation inequality for the fractional power of the Laplacian on Morrey spaces. We then prove a Hardy-type inequality and use it together with the interpolation inequality to obtain a Heisenberg-type inequality in Morrey spaces.

Fractional Hardy-Sobolev inequalities on half spaces

We investigate the existence of extremals for Hardy-Sobolev inequalities involving the Dirichlet fractional Laplacian of order s, 0<s<1, on half-spaces.

Finsler Hardy-Kato's inequality

We prove an improved version of the trace-Hardy inequality, so-called Kato's inequality, on the half-space in Finsler context. The resulting inequality extends the former one obtained by \cite{AFV} in Euclidean context. Also we discuss the validity of the same type of inequalities on open cones.

Extensions of Hardy inequality

In this paper we prove sharp Hardy inequalities by using Maximal function theory. Our results improve and extend the well-known results of G.Hardy \cite{Ha04}, T.Cazenave \cite {Ca03}, J.-Y.Chemin\cite {Ch06} and T.Tao\cite {TT06}.

Critical Hardy inequality on the half-space via the harmonic transplantation

We prove a critical Hardy inequality on the half-space by using the harmonic transplantation. Also we give an improvement of the subcritical Hardy inequality on the half-space, which converges to the critical Hardy inequality. Sobolev type inequalities are also discussed.

半直线上带哈代位势的分数阶薛定谔算子的热核估计

Heat kernel estimates of fractional Schrödinger operators with Hardy potential on half-line

本文研究半直线上带哈代位势的分数阶薛定谔算子，建立了其热核的精确双边估计。

关于分数阶Orlicz边界Hardy不等式

On fractional Orlicz boundary Hardy inequalities

本文研究有界Lipschitz域上的分数阶Orlicz边界Hardy型不等式，并在多类域中对特定临界情形建立了带对数修正的不等式。

The effect of curvature in fractional Hardy--Sobolev inequality with the Spectral Dirichlet Laplacian

We prove the attainability of the best constant in the fractional Hardy--Sobolev inequality with boundary singularity for the Spectral Dirichlet Laplacian. The main assumption is the average concavity of the boundary at the origin.

On the Hardy-Poincaré inequality with boundary singularities

Let $Ø$ be a smooth bounded domain in $\R^N$ with $N\ge 1$. In this paper we study the Hardy-Poincaré inequality with weight function singular at the boundary of $Ø$. In particular we provide sufficient and necessary conditions on the existence of minimizers.

Sharp Hardy-Leray inequality for three-dimensional solenoidal fields with axisymmetric swirl

In this paper, we prove Hardy-Leray inequality for three-dimensional solenoidal (i.e., divergence-free) fields with the best constant. To derive the best constant, we impose the axisymmetric condition only on the swirl components. This partially complements the former work by O. Costin and V. Maz'ya \cite{Costin-Mazya} on the sharp Hardy-Leray inequality for axisymmetric divergence-free fields.

Integral representation using Green function for fractional Hardy equation

Our main aim is to study Green function for the fractional Hardy operator $P:=(-Δ)^s -\fracθ{|x|^{2s}}$ in $\mathbb{R}^N$, where $0<θ<Λ_{N,s}$ and $Λ_{N,s}$ is the best constant in the fractional Hardy inequality. Using Green function, we also show that the integral representation of the weak solution holds.

Asymptotic behavior of extremal functions to an inequality involving Hardy potential and critical Sobolev exponent

In this paper, we study the asymptotic behavior of radial extremal functions to an inequality involving Hardy potential and critical Sobolev exponent. Based on the asymptotic behavior at the origin and the infinity, we shall deduce a strict inequality between two best constants. Finally, as an application of this strict inequality, we consider the existence of nontrivial solution of a quasilinear Brezis-Nirenberg type problem with Hardy potential and critical Sobolev exponent.

Semilinear elliptic equations involving fractional Hardy operators

Our aim in this article is to study semilinear elliptic equations involving a fractional Hardy operator, an absorption and a Radon source in a weighted distributional sense. We show various scenarios, produced by the combined effect of the fractional Hardy potential, the growth of the absorption term and the concentration of the measure, in which existence and uniqueness results holds.

On fractional parabolic equations with Hardy-type potentials

A classification of local asymptotic profiles and strong unique continuation properties are established for a class of fractional heat equations with a Hardy-type potential, via an Almgren-Poon monotonicity formula combined with a blow-up analysis.

On the Sobolev and Hardy constants for the fractional Navier Laplacian

We prove the coincidence of the Sobolev and Hardy constants relative to the "Dirichlet" and "Navier" fractional Laplacians of any real order $m\in(0,\frac{n}{2})$ over bounded domains in $\mathbb R^n$.

Fractional Kirchhoff problem with Hardy potentiel: Breaking of resonance

The present work is concerned with existence of positive solutions for a class of fractional equation involving a Kirchhoff term and singular potential.

Complete classification and nondegeneracy of minimizers for the fractional Hardy-Sobolev inequality, and applications

We study linear and non-linear equations related to the fractional Hardy--Sobolev inequality. We prove nondegeneracy of ground state solutions to the basic equation and investigate existence and qualitative properties, including symmetry of solutions to some perturbed equations.

Self-improvement of pointwise Hardy inequality

We prove the self-improvement of a pointwise $p$-Hardy inequality. The proof relies on maximal function techniques and a characterization of the inequality by curves.

The fractional Hardy inequality with a remainder term

We calculate the regional fractional Laplacian on some power function on an interval. As an application, we prove Hardy inequality with an extra term for the fractional Laplacian on the interval with the optimal constant. As a result, we obtain the fractional Hardy inequality with best constant and an extra lower-order term for general domains, following the method developed by M. Loss and C. Sloane [arXiv:0907.3054v1 [math.AP]]

The second best constant for the Hardy-Sobolev inequality on manifolds

We consider the second best constant in the Hardy-Sobolev inequality on a Riemannian manifold. More precisely, we are interested with the existence of extremal functions for this inequality. This problem was tackled by Djadli-Druet [5] for Sobolev inequalities. Here, we establish the corresponding result for the singular case. In addition, we perform a blow-up analysis of solutions Hardy-Sobolev equations of minimizing type. This yields informations on the value of the second best constant in the related Riemannian functional inequality.

A curl-free improvement of the Rellich-Hardy inequality with weight

We consider the best constant in the Rellich-Hardy inequality (with a radial power weight) for curl-free vector fields on $\mathbb{R}^N$, originally found by Tertikas-Zographopoulos \cite{Tertikas-Z} for unconstrained fields. This inequality is considered as an intermediate between Hardy-Leray and Rellich-Leray inequalities. Under the curl-free condition, we compute the new explicit best constant in the inequality and prove the non-attainability of the constant. This paper is a sequel to \cite{CF_MAAN,CF_Re}.

分数 $p$-拉普拉斯 Hardy-Sobolev 运算符在有界域上的全局紧致性结果

Global compactness results for fractional $p$-Laplace Hardy Sobolev operator on a bounded domain

本文研究了分数 $p$-拉普拉斯 Hardy-Sobolev 运算符在有界域上的非线性临界问题，建立了全局紧致性结果。

A Fractional Hardy-Sobolev-Maz'ya Inequality on the Upper Halfspace

We prove several Sobolev inequalities, which are then used to establish a fractional Hardy-Sobolev- Maz'ya inequality on the upper halfspace.

Quantitative truncation estimates for fractional Hardy-Sobolev optimizers

The general stability problem of truncations for a family of functions concentrating mass at the origin is described and a concrete example in the framework of entire optimizers for the fractional Hardy-Sobolev inequality is given. In this short note we point out some quantitative stability estimates, useful in dealing with critical $p-q$ fractional equations.

New weighted Hardy's inequalities with application to non-existence of global solutions

In this article, we prove a weighted Hardy inequality for $1<p<+\infty$ and dimension $d\ge1$. If $p>d$, then we can deduce from our weighted Hardy inequality a Poincaré inequality. The proof of the weighted Hardy inequality is based on the method of vector fields firstly introduced by Mitidieri \cite{MR1769903}. By the same method, we show for $1<p<+\infty$ and $d\ge1$ that weighted Caffarelli-Kohn-Nirenberg inequalities hold true. As an application of our weighted Hardy inequality, we prove a non-existence result for a p-Kolmogorov parabolic equation.

Localisation for the torsion function and the strong Hardy inequality

Two-sided bounds for the efficiency of the torsion function are obtained in terms of the square of the distance to the boundary function under the hypothesis that the Dirichlet Laplacian satisfies a strong Hardy inequality. Localisation properties of the torsion function are obtained under that hypothesis. An example is analysed in detail.

The heat equation for the Dirichlet fractional Laplacian with Hardy's potentials: properties of minimal solutions and blow-up

Local and global properties of minimal solutions for the heat equation generated by the Dirichlet fractional Laplacian negatively perturbed by Hardy's potentials on open subsets of $\R^d$ are analyzed. As a byproduct we obtain instantaneous blow-up of nonnegative solutions in the supercritical case.

Sobolev and Hardy-Littlewood-Sobolev inequalities: duality and fast diffusion

In the euclidean space, Sobolev and Hardy-Littlewood-Sobolev inequalities can be related by duality. In this paper, we investigate how to relate these inequalities using the flow of a fast diffusion equation in dimension $d\ge3$. The main consequence is an improvement of Sobolev's inequality when $d\ge5$, which involves the various terms of the dual Hardy-Littlewood-Sobolev inequality. In dimension $d=2$, Onofri's inequality plays the role of Sobolev's inequality and can also be related to its dual inequality, the logarithmic Hardy-Littlewood-Sobolev inequality, by a super-fast diffusion equat…

On the integral systems related to Hardy-Littlewood-Sobolev inequality

We prove all the maximizers of the sharp Hardy-Littlewood-Sobolev inequality are smooth. More generally, we show all the nonnegative critical functions are smooth, radial with respect to some points and strictly decreasing in the radial direction. In particular, this resolves all the cases left open by previous works of Chen, Li and Ou on the corresponding integral systems.

Sharp nonexistence results for a linear elliptic inequality involving Hardy and Leray potentials

In this paper we deal with nonnegative distributional supersolutions for a class of linear elliptic equations involving inverse-square potentials and logarithmic weights. We prove sharp nonexistence results.

齐次Lie群上 $Q<sp$ 情形下的分数阶Hardy型不等式

Fractional Hardy type inequalities on homogeneous Lie groups in the case $Q<sp$

本文研究了齐次Lie群上 $Q<sp$ 互补情形下的分数阶Hardy不等式及其不确定性原理，并进一步证明了分数阶Hardy-Sobolev型、对数Hardy-Sobolev型及Nash型不等式。

关于带Hardy型势的Schrödinger算子的分数次幂

On the fractional powers of a Schrödinger operator with a Hardy-type potential

该研究针对带Hardy型势的Schrödinger算子分数次幂，结合Almgren单调性公式与爆破分析，建立强唯一延拓性质并完成渐近轮廓分类。

Improvements and generalizations of two Hardy type inequalities and their applications to the Rellich type inequalities

We give improvements and generalizations of both the classical Hardy inequality and the geometric Hardy inequality based on the divergence theorem. Especially, our improved Hardy type inequality derives both two Hardy type inequalities with best constants. Besides, we improve two Rellich type inequalities by using the improved Hardy type inequality.

Weighted Hardy inequality with higher dimensional singularity on the boundary

Let $Ω$ be a smooth bounded domain in $\mahbb R^N$ with $N\ge 3$ and let $Σ_k$ be a closed smooth submanifold of $δΩ$ of dimension $1\le k\le N-2$. In this paper we study the weighted Hardy inequality with weight function singular on $Σ_k$. In particular we provide sufficient and necessary conditions for existence of minimizers.

A Hardy-Littlewood-Sobolev type inequality for variable exponents and applications to quasilinear Choquard equations involving variable exponent

In this work, we have proved a version of the Hardy-Littlewood-Sobolev inequality for variable exponents. After we use the variational method to establish the existence of solution for a class of Choquard equations involving the $p(x)$-Laplacian operator.

Hardy's inequality for functions vanishing on a part of the boundary

We develop a geometric framework for Hardy's inequality on a bounded domain when the functions do vanish only on a closed portion of the boundary.

On Bilinear Hardy Inequality and Corresponding Geometric Mean Inequality

The main aim of this paper to provide several scales of equivalent conditions for the bilinear Hardy inequalities in the case $1< q, p_1, p_2<\infty$ with $q \geq \max(p_1,p_2)$.

Fractional Poincaré and localized Hardy inequalities on metric spaces

We prove fractional Sobolev-Poincaré inequalities, capacitary versions of fractional Poincaré inequalities, and pointwise and localized fractional Hardy inequalities in a metric space equipped with a doubling measure. Our results generalize and extend earlier work where such inequalities have been considered in the Euclidean spaces or in the non-fractional setting in metric spaces. The results concerning pointwise and localized variants of fractional Hardy inequalities are new even in the Euclidean case.

带权和共形分数阶微分算子的各向异性Hardy型不等式

Anisotropic Hardy type inequalities with weights and conformable fractional differential operators

该研究通过发展共形微积分基本概念建立相关定理与恒等式，结合新各向异性Picone型恒等式，推导出带权和共形分数阶微分算子的广义各向异性Hardy型不等式，并得到若干Hardy型不等式及海森堡-泡利-外尔不确定性原理。