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.
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 $\mathbb{H}^n$.
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.
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.
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 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.
An extension of a critical Hardy--Rellich inequality: explicit constants and the sharp weight range
Imam Abdulrahman Bin Faisal University（伊玛姆阿卜杜勒拉赫曼·本·费萨尔大学）；Basic and Applied Scientific Research Center, Imam Abdulrahman Bin Faisal University（伊玛姆阿卜杜勒拉赫曼·本·费萨尔大学基础与应用科学研究中心）
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.
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.
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.
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.
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.
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}.
Extremizer Stability of Higher-order Hardy-Rellich inequalities for Baouendi--Grushin vector fields
Indian Statistical Institute, Delhi Center（印度统计学院德里中心）；National Taiwan Normal University（台湾师范大学）；Indian Institute of Technology Hyderabad（印度理工学院海得拉巴分校）
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.
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.
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.
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…
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…
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.
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…
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-…
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…
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 …
Hardy and Rellich identities and inequalities for Baouendi-Grushin operators via spherical vector fields
Indian Statistical Institute（印度统计研究所）；Indian Institute of Science Education and Research Mohali（印度科学教育与研究学院莫哈利分校）；Indian Institute of Technology Hyderabad（印度理工学院海得拉巴分校）
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 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…
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 $C_0^{\infty}(\mathbb{R}^n \backslash \{0\})$…
In this note we prove horizontal weighted Rellich inequalities for the sub-Laplacian and for sub-Laplacians with drift on general stratified groups. We show how the presence of a drift improves the known inequalities. Moreover, we obtain several versions of weighted Rellich inequalities for the sub-Laplacian with drift on the polarizable Carnot groups, also with the weights associated with the fundamental solution of the sub-Laplacian. The obtained results are already new for the Laplacian in the usual Euclidean setting of ${\mathbb R}^n$, embedding the classical Rellich inequality into a fami…
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.
The purpose of this article is to study extrapolation of solvability for boundary value problems of elliptic systems in divergence form on the upper half-space assuming De Giorgi type conditions. We develop a method allowing to treat each boundary value problem independently of the others. We shall base our study on solvability for energy solutions, estimates for boundary layers, equivalence of certain boundary estimates with interior control so that solvability reduces to a one-sided Rellich inequality. Our method then amounts to extrapolating this Rellich inequality using atomic Hardy spaces…
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 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.
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.
Sharp extensions of Pitt's inequality and bounds for Stein-Weiss fractional integrals are obtained that incorporate gradient forms and vector-valued operators. Such results include Hardy-Rellich inequalities.
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…
