A Sequence of Weighted Birman-Hardy-Rellich Inequalities with Logarithmic Refinements

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.

Hardy, Rellich and Uncertainty principle inequalities on Carnot Groups

In this paper we prove sharp weighted Hardy-type inequalities on Carnot groups with the homogeneous norm $N=u^{1/(2-Q)}$ associated to Folland's fundamental solution $u$ for the sub-Laplacian $Δ_{\mathbb{G}}$. We also prove uncertainty principle, Caffarelli-Kohn-Nirenberg and Rellich inequalities on Carnot groups.

一维积分型 Rellich 不等式

One-dimensional integral Rellich type inequalities

本文受新型 Hardy 不等式启发，研究一维积分形式的 Hardy-Rellich 与 Rellich 不等式，改进了已有尖锐 Hardy-Rellich 型结果，并给出新的尖锐 Rellich 型积分不等式。

关于Baouendi - Grushin向量场的高阶Hardy - Rellich不等式的极值稳定性

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。

Sharp Poincaré-Hardy and Poincaré-Rellich inequalities on the hyperbolic space

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 …

关于Hardy、Copson和Rellich不等式的改进

On improvements of the Hardy, Copson and Rellich inequalities

本文通过因式分解方法和广义离散Dirichlet拉普拉斯矩阵，建立并改进了一维离散Hardy、Rellich和Copson不等式，证明了改进权重的最佳性，并揭示了Knopp不等式与Rellich不等式改进之间的依赖关系。

半直线上的改进型离散Rellich不等式

An improved discrete Rellich inequality on the half-line

该研究采用因式分解新思路证明半直线上的改进型离散Rellich不等式并讨论其最优性，同时提出高阶离散类Hardy不等式的相关猜想与全空间离散Rellich不等式的开放问题。

具有最优Hardy-Rellich不等式和临界指数的Sobolev插值不等式

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。

The Hardy-Rellich inequality and uncertainty principle on the sphere

Let $Δ_0$ be the Laplace-Beltrami operator on the unit sphere $\mathbb{S}^{d-1}$ of $\mathbb{R}^d$. We show that the Hardy-Rellich inequality of the form $$ \int_{\mathbb{S}^{d-1}} \left | f (x)\right|^2 dσ(x) \leq c_d \min_{e\in \mathbb{S}^{d-1}} \int_{\mathbb{S}^{d-1}} (1- \langle x, e \rangle) \left |(-Δ_0)^{\frac{1}{2}}f(x) \right |^2 dσ(x) $$ holds for $d =2$ and $d \ge 4$ but does not hold for $d=3$ with any finite constant, and the optimal constant for the inequality is $c_d = 8/(d-3)^2$ for $d =2, 4, 5$ and, under additional restrictions on the function space, for $d\ge 6$. This inequa…

半直线上等式框架下的Rellich型不等式

Rellich type inequalities on the half line in the framework of equalities

本文在等式框架下建立了半直线上的加权Hardy和Rellich型不等式，推广了近期相关结果，并证明了一维经典Rellich不等式在等式框架下成立。

On Weighted Hardy-Type Inequalities

We revisit weighted Hardy-type inequalities employing an elementary ad hoc approach that yields explicit constants. We also discuss the infinite sequence of power weighted Birman-Hardy-Rellich-type inequalities and derive an operator-valued version thereof.

An improved bound for the non-existence of radial solutions of the Brezis-Nirenberg problem in Hn

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.

Improved higher order Poincaré inequalities on the hyperbolic space via Hardy-type remainder terms

The paper deals about Hardy-type inequalities associated with the following higher order Poincaré inequality: $$ \left( \frac{N-1}{2} \right)^{2(k -l)} := \inf_{ u \in C_{c}^{\infty} \setminus \{0\}} \frac{\int_{\mathbb{H}^{N}} |\nabla_{\mathbb{H}^{N}}^{k} u|^2 \ dv_{\mathbb{H}^{N}}}{\int_{\mathbb{H}^{N}} |\nabla_{\mathbb{H}^{N}}^{l} u|^2 \ dv_{\mathbb{H}^{N}} }\,, $$ where $0 \leq l < k$ are integers and $\mathbb{H}^{N}$ denotes the hyperbolic space. More precisely, we improve the Poincaré inequality associated with the above ratio by showing the existence of $k$ Hardy-type remainder terms. F…

半直线上的最优分数阶离散Hardy不等式

Optimal fractional discrete Hardy inequalities on the half-line

该研究将半直线上分数阶离散拉普拉斯算子的Toeplitz实现视为全空间算子的压缩，证明了强最优的分数阶Hardy不等式，且等号无法由非零元达到，进而得到离散Birman不等式的分数阶推广。