The Hardy-Rellich Inequality for Polyharmonic Operators

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

The principal aim of this paper is to extend Birman's sequence of integral inequalities originally obtained in 1961, and containing Hardy's and Rellich's inequality as special cases, to a sequence of inequalities that incorporates power weights on either side and logarithmic refinements on the right-hand side of the inequality as well. Our new technique of proof for this sequence of inequalities relies on a combination of transforms originally due to Hartman and Müller-Pfeiffer. The results obtained considerably improve on prior results in the literature.

最优离散Hardy-Rellich-Birman不等式

Optimal discrete Hardy-Rellich-Birman inequalities

本研究给出半线上离散拉普拉斯整数次幂不等式中确定最优权的参数序列充分条件，通过具体选取得到显式最优离散Rellich与Birman权，改进了已有相关结果并证明了一则猜想。

On Birman's sequence of Hardy-Rellich-type inequalities

In 1961, Birman proved a sequence of inequalities $\{I_{n}\},$ for $n\in\mathbb{N},$ valid for functions in $C_0^{n}((0,\infty))\subset L^{2}((0,\infty)).$ In particular, $I_{1}$ is the classical (integral) Hardy inequality and $I_{2}$ is the well-known Rellich inequality. In this paper, we give a proof of this sequence of inequalities valid on a certain Hilbert space $H_{n}([0,\infty))$ of functions defined on $[0,\infty).$ Moreover, $f\in H_{n}([0,\infty))$ implies $f^{\prime}\in H_{n-1}([0,\infty));$ as a consequence of this inclusion, we see that the classical Hardy inequality implies each…

在欧几里得和非欧几里得环境下统一的 $L^p$ Hardy 和 Rellich 型不等式方法

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。

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。

On families of optimal Hardy-weights for linear second-order elliptic operators

We construct families of optimal Hardy-weights for a subcritical linear second-order elliptic operator using a one-dimensional reduction. More precisely, we first characterise all optimal Hardy-weights with respect to one-dimensional subcritical Sturm-Liouville operators on a given interval, and then apply this result to obtain families of optimal Hardy inequalities for general linear second-order elliptic operators in higher dimensions. As an application, we prove a new Rellich inequality.

Hypoelliptic functional inequalities

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…

Lieb-Thirring Inequalities for Fourth-Order Operators in Low Dimensions

This paper considers Lieb-Thirring inequalities for higher order differential operators. A result for general fourth-order operators on the half-line is developed, and the trace inequality tr((-Delta)^2 - C^{HR}_{d,2} / (|x|^4) - V(x))^{-γ} < C_γ\int_{R^d} V(x)_+^{γ+ d/4} dx for gamma \geq 1 - d/4, where C^{HR}_{d,2} is the sharp constant in the Hardy-Rellich inequality and where C_γ> 0 is independent of V, is proved for dimensions d = 1,3. As a corollary of this inequality a Sobolev-type inequality is obtained.