Cristian Cazacu
In this survey we give a compact presentation of well-known functional inequalities of Hardy and Rellich type in the $L^2$ setting. In addition, we give some insights of their proofs by using standard and basic tools such as the method of super-solutions.
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In this paper, we establish the weighted anisotropic Hardy and Rellich type inequalities with boundary terms for general (real-valued) vector fields. As consequences, we derive new as well as many of the fundamental Hardy and Rellich type inequalities which are known in different settings.
In this paper, we prove Hardy-Leray and Rellich-Leray inequalities for curl-free vector fields with sharp constants. This complements the former work by Costin-Maz'ya \cite{Costin-Mazya} on the sharp Hardy-Leray inequality for axisymmetric divergence-free vector fields.
This paper is primarily devoted to a class of interpolation inequalities of Hardy and Rellich types on the Heisenberg group $\mathbb{H}^n$. Consequently, several weighted Hardy type, Heisenberg-Pauli-Weyl uncertainty principle and Hardy-Rellich type inqualities are established on $\mathbb{H}^n$. Moreover, new weighted Sobolev type embeddings are derived. Finally, an integral inequality for vector fields in a domain of the Heisenberg group is obtained, leading to several specific weighted Hardy type inequalities by making careful choices of vector fields.
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。
First the Hardy and Rellich inequalities are defined for the submarkovian operator associated with a local Dirichlet form. Secondly, two general conditions are derived which are sufficient to deduce the Rellich inequality from the Hardy inequality. In addition the Rellich constant is calculated from the Hardy constant. Thirdly, we establish that the criteria for the Rellich inequality are verified for a large class of weighted second-order operators on a domain $Ω\subseteq \Ri^d$. The weighting near the boundary $\partial Ω$ can be different from the weighting at infinity. Finally these result…
We prove a family of Hardy-Rellich and Poincaré identities and inequalities on the hyperbolic space having, as particular cases, improved Hardy-Rellich, Rellich and second order Poincaré inequalities. All remainder terms provided considerably improve those already known in literature, and all identities hold with same constants for radial operators also. Furthermore, as applications of the main results, second order versions of the uncertainty principle on the hyperbolic space are derived.
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}.
Fractional Hardy-Rellich inequalities via integration by parts
本研究在$C^{1,1}$类有界区域中证明带显式常数的分数阶Hardy-Rellich不等式，通过依托Pohozaev型恒等式推广了E. Mitidieri提出的经典证明方法。
In this paper we establish the subelliptic Picone type identities. As consequences, we obtain Hardy and Rellich type inequalities for anisotropic p-sub- Laplacians which are operators of the form $$ \mathcal{L}_p f := \sum_{i=1}^{N}X_i(|X_if|^{p_i-2}X_i f), \quad 1<p_i< \infty, $$ where $X_i, i=1,\ldots, N,$ are the generators of the first stratum of a stratified (Lie) group. Moreover, analogues of Hardy type inequalities with multiple singularities and many-particle Hardy type inequalities are obtained on stratified groups.
In this paper we obtain logarithmic Hardy and Rellich inequalities on general Lie groups. In the case of graded groups, we also show their refinements using the homogeneous Sobolev norms. In fact, we derive a family of weighted logarithmic Hardy-Rellich inequalities, for which logarithmic Hardy and Rellich inequalities are special cases. As a consequence of these inequalities, we also derive a Gross type logarithmic Hardy inequality on general stratified groups. An interesting feature of such estimate is that we consider the measure which is Gaussian only on the first stratum of the group. Suc…
We will present versions of the Rellich-Kondrachov theorem for pseudo-differential operators acting on localizable Hardy spaces. One of the techniques includes boundedness properties for pseudodifferential operators with symbols in the Hörmander class, that might be of interest in his own right, extending results previously obtained by Goldberg, Alvarez and Hounie, Taylor - among others. As application, we obtain compact embedding results for distributions in nonhomogeneous localizable Hardy-Sobolev spaces.
Higher order Hardy-Rellich identities
本文建立了带Hardy-Hénon权重的多调和与径向Laplacian算子的高阶Hardy-Rellich恒等式，并推广至黎曼流形，进而改进和扩展了Rellich不等式，包括双曲空间上的新结果。
Weighted Hardy-Rellich type inequalities: improved best constants and symmetry breaking
该研究针对L²空间中径向导数替代全梯度的加权Hardy-Rellich不等式，确定了新的精确常数，证明部分权重下常数严格更大，且部分情形中极值函数无径向对称性。
The Hardy-Rellich inequality in the whole space with the best constant was firstly proved by Tertikas and Zographopoulos in Adv. Math. (2007) in higher dimensions $N\geq 5$. Then it was extended to lower dimensions $N\in \{3, 4\}$ by Beckner in Forum Math. (2008) and Ghoussoub-Moradifam in Math. Ann. (2011) by applying totally different techniques. In this note we refine the method implemented by Tertikas and Zographopoulos, based on spherical harmonics decomposition, to give an easy and compact proof of the optimal Hardy-Rellich inequality in any dimension $N\geq 3$. In addition, we provide m…
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 $\mathbb{H}^n$.
Double Criticality for a Hardy-Rellich Biharmonic Heat Equation in an Exterior Domain
研究外域上含奇异Hardy-Rellich势、加权非线性和正源项的双调和热方程弱解的存在性与非存在性，识别两种不同临界情形，提出Fujita型临界指数及源项衰减指数的第二临界指数，扩展了Tobakhanov近期工作。
We establish existence of weighted Hardy and Rellich inequalities on the spaces $L_p(Ω)$ where $Ω= \Ri^d\backslash K$ with $K$ a closed convex subset of $\Ri^d$. Let $Γ=\partialΩ$ denote the boundary of $Ω$ and $d_Γ$ the Euclidean distance to $Γ$. We consider weighting functions $c_Ω=c\circ d_Γ$ with $c(s)=s^δ(1+s)^{δ'-δ}$ and $δ,δ'\geq0$. Then the Hardy inequalities take the form \[ \int_Ωc_Ω\,|\nablaφ|^p\geq b_p\int_Ωc_Ω\,d_Γ^{\;-p}\,|φ|^p \] and the Rellich inequalities are given by \[ \int_Ω|Hφ|^p\geq d_p\int_Ω|c_Ω\,d_Γ^{\,-2}φ|^p \] with $H=-\divv(c_Ω\nabla)$. The constants $b_p, d_p$ dep…
One dimensional sharp discrete Hardy-Rellich inequalities
本文在自然数集上建立了带最优常数的离散Hardy-Rellich不等式，通过加权等式与迭代方法处理高阶情形，并推广到一般图及$\ell^p$空间。
A new approach to weighted Hardy-Rellich inequalities: improvements, symmetrization principle and symmetry breaking
本研究给出加权Hardy-Rellich不等式成立的权重充要条件，提出用Bessel对建立最优不等式的新方法，改进了已有结果，并部分解答了相关公开问题，明确了不等式对所有函数成立等价于对径向函数成立的权重条件。
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.
Sharp remainder formulae for general weighted Hardy and Rellich type inequalities for $1<p<\infty$
本文提出了一般加权Hardy和Rellich型不等式中精确余项的公式，扩展了原有结果并揭示了其在经典拉普拉斯算子上的新应用。
We give necessary and sufficient conditions on a pair of positive radial functions V and W on a ball B of radius R in R^n,$n \geq 1$, so that the following inequalities hold for all $u \in C_{0}^{\infty}(B)$: $\int_{B}V(x)|\nabla u |^{2}dx \geq \int_{B} W(x)u^2dx$, and $\int_{B}V(x)|Δu |^{2}dx \geq \int_{B} W(x)|\nabla u|^{2}dx+(n-1)\int_{B}(\frac{V(x)}{|x|^2}-\frac{V_r(|x|)}{|x|})|\nabla u|^2dx$. This characterization makes a very useful connection between Hardy-type inequalities and the oscillatory behaviour of certain ordinary differential equations, and helps in the identification of a lar…
Hardy and Rellich identities and inequalities for Baouendi-Grushin operators via spherical vector fields
本研究针对Baouendi-Grushin向量场，结合Bessel对抽象恒等式、球谐函数与构造的球面向量场，证明了带最优常数和显式余项的Hardy、Rellich类恒等式与不等式，推导了新的不确定性原理与对称化原理。
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.
Asymptotically sharp Hardy-Rellich inequalities on lattices
研究格点\(\mathbb{Z}^d\)上离散哈代 - 雷利奇不等式最优常数的渐近行为，通过傅里叶约简、哈代 - 雷利奇恒等式及概率集中估计等方法，证明\(\lim_{d\to\infty}\frac{\mathcal C_{m,d}}{d^m}=2^m\)。
Factorizations and Power Weighted Rellich and Hardy--Rellich-Type Inequalities
本文重新审视并扩展了与幂加权 Rellich 及 Hardy--Rellich 不等式相关的多种不等式，包括 Schmincke 提出的不等式。
Some of the most known integral inequalities are the Sobolev, Hardy and Rellich inequalities in Euclidean spaces. In the context of submanifolds, the Sobolev inequality was proved by Michael-Simon and Hoffman-Spruck. Since then, a sort of applications to the submanifold theory has been derived from those inequalities. Years later, Carron obtained a Hardy inequality for submanifolds in Hadamard spaces. In this paper, we prove the general Hardy and Rellich Inequalities for submanifolds in Hadamard spaces. Some applications are given and we also analyse the equality cases.
In this article, we look for the weight functions (say $g$) that admits the following generalized Hardy-Rellich type inequality: $ \int_Ω g(x) u^2 dx \leq C \int_Ω |Δu|^2 dx, \forall u \in \mathcal{D}^{2,2}_0(Ω), $ for some constant $C>0$, where $Ω$ is an open set in $\mathbb{R}^N$ with $N\ge 1$. We find various classes of such weight functions, depending on the dimension $N$ and the geometry of $Ω.$ Firstly, we use the Muckenhoupt condition for the one dimensional weighted Hardy inequalities and a symmetrization inequality to obtain admissible weights in certain Lorentz-Zygmund spaces. Second…
We consider Hardy-Rellich inequalities and discuss their possible improvement. The procedure is based on decomposition into spherical harmonics, where in addition various new inequalities are obtained (e.g. Rellich-Sobolev inequalities). We discuss also the optimality of these inequalities in the sense that we establish (in most cases) that the constants appearing there are the best ones. Next, we investigate the polyharmonic operator (Rellich and Higher Order Rellich inequalities); the difficulties arising in this case come from the fact that (generally) minimizing sequences are no longer exp…
We give necessary and sufficient conditions on a pair of positive radial functions $V$ and $W$ on a ball $B$ of radius $R$ in $R^{n}$, $n \geq 1$, so that the following inequalities hold \begin{equation*} \label{two} \hbox{$\int_{B}V(x)|\nabla u |^{2}dx \geq \int_{B} W(x) u^{2}dx+b\int_{\partial B}u^2 ds$ for all u $\in H^1(B)$,} \end{equation*} and \begin{equation*} \label{two} \hbox{$\int_{B}V(x)|Δu |^{2}dx \geq \int_{B} W(x)|\nabla u|^{2}dx+b\int_{\partial B}|\nabla u|^2 ds$ for all u $\in H^2(B)$.} \end{equation*} Then we present various classes of optimal weighted Hardy-Rellich inequaliti…
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。
In this paper we study Hardy and Rellich type inequalities for Baouendi-Grushin vector fields : $\nabla_γ=(\nabla_x, |x|^{2γ}\nabla_y)$ where $γ>0$, $\nabla_x$ and $\nabla_y$ are usual gradient operators in the variables $x\in \mathbb{R}^m$ and $y\in\mathbb{R}^k$, respectively. In the first part of the paper, we prove some weighted Hardy type inequalities with remainder terms. In the second part, we prove two versions of weighted Rellich type inequality on the whole space. We find sharp constants for these inequalities. We also obtain their improved versions for bounded domains.
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.
We prove local refined versions of Hardy's and Rellich's inequalities as well as of uncertainty principles for sums of squares of vector fields on bounded sets of smooth manifolds under certain assumptions on the vector fields. We also give some explicit examples, in particular, for sums of squares of vector fields on Euclidean spaces and for sub-Laplacians on stratified Lie 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.
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.
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.
Logarithmic Refinements of a Power Weighted Hardy--Rellich-Type Inequality
本文证明了在$n$维球上幂加权Hardy--Rellich不等式的一个对数加细，该结果适用于最大范围的底层参数及所有维度$n \geq 2$。
We study Hardy-type inequalities associated to the quadratic form of the shifted Laplacian $-Δ_{\mathbb H^N}-(N-1)^2/4$ on the hyperbolic space ${\mathbb H}^N$, $(N-1)^2/4$ being, as it is well-known, the bottom of the $L^2$-spectrum of $-Δ_{\mathbb H^N}$. We find the optimal constant in the resulting Poincaré-Hardy inequality, which includes a further remainder term which makes it sharp also locally. A related inequality under suitable curvature assumption on more general manifolds is also shown. Similarly, we prove Rellich-type inequalities associated with the shifted Laplacian, in which at …
The principal aim of this note is to illustrate how factorizations of singular, even-order partial differential operators yield an elementary approach to classical inequalities of Hardy-Rellich-type. More precisly, introducing the two-parameter $n$-dimensional homogeneous scalar differential expressions $T_{α,β} := - Δ+ α|x|^{-2} x \cdot \nabla + β|x|^{-2}$, $α, β\in \mathbb{R}$, $x \in \mathbb{R}^n \backslash \{0\}$, $n \in \mathbb{N}$, $n \geq 2$, and its formal adjoint, denoted by $T_{α,β}^+$, we show that nonnegativity of $T_{α,β}^+ T_{α,β}$ on $C_0^{\infty}(\mathbb{R}^n \backslash \{0\})$…
We investigate Hardy-Rellich inequalities for perturbed Laplacians. In particular, we show that a non-trivial angular perturbation of the free operator typically improves the inequality, and may also provide an estimate which does not hold in the free case. The main examples are related to the introduction of a magnetic field: this is a manifestation of the diamagnetic phenomenon, which has been observed by Laptev and Weidl in \cite{LW1999} for the Hardy inequality, later by Evans and Lewis in \cite{EL2005} for the Rellich inequality; however, to the best of our knowledge, the so called Hardy-…
An extension of a critical Hardy--Rellich inequality: explicit constants and the sharp weight range
本文通过Emden-Fowler变量中的经典一维加权Hardy不等式，将临界Hardy-Rellich不等式推广到$a\neq N$的情形，推导了显式常数并确定了尖锐权值范围。
Extremizer Stability of Higher-order Hardy-Rellich inequalities for Baouendi--Grushin vector fields
研究在径向Baouendi - Grushin向量场设定下的\(L^p\)-Rellich和Hardy - Rellich不等式，通过建立恒等式、分析极值稳定性改进不等式，推导高阶不等式并证明常数精确，计算\(L^2\)高阶情形的精确余项。
A critical Hardy-Rellich inequality
本文证明了临界Hardy-Rellich型不等式，针对N≥1，证明了对于任意u∈C^∞_c(R^N\{0}），存在常数C_N>0，使得积分不等式成立。
A unified approach to $L^p$ Hardy and Rellich-type inequalities in Euclidean and non-Euclidean settings
本文提出一种统一的方法，用于建立广泛子椭圆算子的 $L^p$ Hardy 和 Rellich 不等式，适用于欧几里得和非欧几里得环境，并给出精确常数。
Sobolev interpolation inequalities with optimal Hardy-Rellich inequalities and critical exponents
本文建立了一族新的临界高阶Sobolev插值不等式，覆盖径向与非径向函数，利用尖锐Hardy-Rellich不等式的最优二次型，并通过Fourier分析和高阶Talenti比较原理克服高阶情形下的困难，推广了Dietze和Nam的一阶结果。
Rellich inequalities via Riccati pairs on model space forms
本文利用凸性论证、分部积分和基于Riccati型常微分不等式可解性的Riccati对，在常值非正截面曲率黎曼流形上建立了Rellich不等式，推广了已有Hardy不等式结果。
Hardy and Rellich inequalities with Bessel pairs
本文利用 Bessel 对在有界区域上刻画了函数对满足 Hardy 与 Rellich 不等式的条件，给出 $L^p$ 推广及多种算子实例，并解决了 Ghoussoub-Moradifam 的开放问题。
Stability of Rellich-Sobolev type inequality involving Hardy term for bi-Laplacian
本文证明了涉及双调和算子Hardy项的Rellich-Sobolev型不等式极值函数的非退化结果，通过谱分析与紧性论证研究其稳定性，并在有限Lebesgue测度子域中导出弱Lebesgue范数意义下的余项不等式。
We prove two improved versions of the Hardy-Rellich inequality for the polyharmonic operator $(-Δ)^m$ involving the distance to the boundary. The first involves an infinite series improvement using logarithmic functions, while the second contains $L^2$ norms and involves as a coefficient the volume of the domain. We find explicit constants for these inequalities, and we prove their optimality in the first case.
On Rellich-type asymptotics for eigenfunctions on rank one symmetric spaces of noncompact type
研究非紧秩为一黎曼对称空间上拉普拉斯-贝尔特拉米算子的特征函数在外部领域的行为，建立精确的L^p增长估计，揭示非欧几里得谱现象。
The classical Hardy inequality holds in Sobolev spaces $W_0^{1,p}$ when $1\le p< N$. In the limiting case where $p=N$, it is known that by adding a logarithmic function to the Hardy potential, some inequality which is called the critical Hardy inequality holds in $W_0^{1,N}$. In this note, in order to give an explanation of appearance of the logarithmic function at the potential, we derive the logarithmic function from the classical Hardy inequality with the best constant via some limiting procedure as $p \nearrow N$. And we show that our limiting procedure is also available for the classical …
We prove \emph{optimal} improvements of the Hardy inequality on the hyperbolic space. Here, optimal means that the resulting operator is critical in the sense of [J.Funct.Anal. 266 (2014), pp. 4422-89], namely the associated inequality cannot be further improved. Such inequalities arise from more general, \emph{optimal} ones valid for the operator $ P_λ:= -Δ_{\mathbb{H}^N} - λ$ where $0 \leq λ\leq λ_{1}(\mathbb{H}^N)$ and $λ_{1}(\mathbb{H}^N)$ is the bottom of the $L^2$ spectrum of $-Δ_{\mathbb{H}^N} $, a problem that had been studied in [J.Funct.Anal. 272 (2017), pp. 1661-1703 ] only for the …
On the Stein-Weiss inequalities and higher-order Caffarelli-Kohn-Nirenberg type inequalities: sharp constants, symmetry of extremal functions
本文通过分类加权四阶方程的径向对称解，推导了多类带最佳常数与径向对称极值函数的加权不等式，包括Stein-Weiss、Caffarelli-Kohn-Nirenberg型及Hardy-Rellich等不等式。
On the Optimality and Decay of $p$-Hardy Weights on Graphs
该研究针对局部有限图上拟线性Schrödinger算子的次临界能量泛函构造最优$p$-Hardy权重，证明其偏序最大与零临界性，给出权重衰减条件并推导零临界性蕴含无穷远最优性，还讨论了相关不等式与原理。
Sharp second order inequalities with distance function to the boundary and applications to a p-Biharmonic singular problem
本文将带边界距离函数奇异项的Hardy-Rellich不等式推广到L^p情形，给出尖锐常数或新界，并应用于p-双调和奇异问题解的存在性与非存在性研究。
Using a Rellich-Pohozaev argument and Hardy's inequality, we derive an improved bound on the nonlinear eigenvalue for the non existence of radial solutions of a Brezis-Nirenberg problem, with Dirichlet boundary conditions, on a geodesic ball of Hn, for 2<n<4.
In this paper we derive a variety of functional inequalities for general homogeneous invariant hypoelliptic differential operators on nilpotent Lie groups. The obtained inequalities include Hardy, Rellich, Hardy-Littllewood-Sobolev, Galiardo-Nirenberg, Caffarelli-Kohn-Nirenberg and Trudinger-Moser inequalities. Some of these estimates have been known in the case of the sub-Laplacians, however, for more general hypoelliptic operators almost all of them appear to be new as no approaches for obtaining such estimates have been available. Moreover, we obtain several versions of local and global wei…
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