On first order interpolation inequalities with weights on the Heisenberg group

In this paper, sufficient and necessary conditions for the first order interpolation inequalities with weights on the Heisenberg group are given. The necessity is discussed by polar coordinates changes of the Heisenberg group. Establishing a class of Hardy type inequalities via a new representation formula for functions and Hardy-Sobolev type inequalities by interpolation, we derive the sufficiency. Finally, sharp constants for Hardy type inequalities are determined.     

A simple approach to Hardy inequalities

We describe a simple method of proving Hardy-type inequalities of second and higher order with weights for functions defined in ℝ ⁿ . It is shown that we can obtain such inequalities with sharp constants by applying the divergence theorem to specially chosen vector fields. Another approach to Hardy inequalities based on the application of identities of Rellich-Pokhozhaev type is also proposed. Translated fromMatematicheskie Zametki, Vol. 67, No. 4, pp. 563–572, April, 2000.     

海森堡群上半空间内最佳Hardy不等式

证明了一类海森堡群上半空间内与次拉普拉斯算子相关的最佳Hardy不等式.作为应用,我们得到了相应的最佳Rellich型不等式.     

Weighted inequalities for higher dimensional one-sided Hardy–Littlewood maximal function in Orlicz spaces

In this paper, weighted extra-weak and weak type inequalities have been characterized for the one-sided Hardy–Littlewood maximal function on the plane. We have addressed conditions on pair of weights for which the dyadic one-sided maximal function on higher dimension is locally integrable. In the process, we characterize weights for which the one-sided Hardy-Littlewood maximal function satisfies restricted weak type inequalities on the plane, thus extending the result of Kerman and Torchinsky to the one-sided Hardy-Littlewood maximal function.     

Dynamic Picone's identity and its applications

We prove some Picone-type identities and inequalities for a class of first-order nonlinear dynamic systems and derive various weighted inequalities of Wirtinger type and Hardy type on time scales. As applications we study oscillatory and related properties of these systems including Reid's roundabout theorem on disconjugacy, Sturm's separation and comparison theorems, as well as a variational method in the oscillation theory.     

Finsler Hardy inequalities

In this paper we present a unified simple approach to anisotropic Hardy inequalities in various settings. We consider Hardy inequalities which involve a Finsler distance from a point or from the boundary of a domain. The sharpness and the non-attainability of the constants in the inequalities are also proved.     

HEISENBERG-GREINER p-退化椭圆算子的一类HARDY不等式及其应用

本文研究了p-退化椭圆Heisenberg-Greiner算子的一类Hardy不等式的证明问题.利用正则化及逼近的方法,获得了该算子的一类强Hardy不等式的结果,推广了已有文献中p的取值范围,并进一步弥补了去除原点的缺陷.作为应用,讨论了与该算子相关的一类非线性算子的正定性与下无界性,并给出了它的一个正解.     

Some Hardy Type Inequalities in R~n

Some Hardy type inequalities on the ball and its complementary set in the Euclidean space are established by using the Picone type identity and constructing suitable auxiliary functions.     

Heisenberg型群上的几类Hardy型不等式

文章得到了Heisenberg型群上的几类Hardy型不等式,并确定出了次Laplace算子的Hardy型不等式中的最佳常数．     

Hardy and Rellich-Type Inequalities for Metrics Defined by Vector Fields

Let {X _i }, i=1,...,m be a system of locally Lipschitz vector fields on DR ⁿ , such that the corresponding intrinsic metric is well-defined and continuous w.r.t. the Euclidean topology. Suppose that the Lebesgue measure is doubling w.r.t. the intrinsic balls, that a scaled L¹ Poincaré inequality holds for the vector fields at hand (thus including the case of Hörmander vector fields) and that the local homogeneous dimension near a point x ₀ is sufficiently large. Then weighted Sobolev–Poincaré inequalities with weights given by power of (,x ₀) hold; as particular cases, they yield non-local analogues of both Hardy and Sobolev–Okikiolu inequalities. A general argument which shows how to deduce Rellich-type inequalities from Hardy inequalities is then given: this yields new Rellich inequalities on manifolds and even in the uniformly elliptic case. Finally, applications of Sobolev–Okikiolu inequalities to heat kernel estimates for degenerate subelliptic operators and to criteria for the absence of bound states for Schrödinger operators H=–L+V are given.     

与广义Baouendi-Grushin向量场相联系的Hardy不等式及其应用

本文建立一类与广义Baouendi—Grushin向量场联系的Hardy不等式．采用的技巧是延伸欧氏空间上的散度定理推出的基本积分不等式和选定适当的向量场．Hardy不等式相应的最佳常数也得到证明．本文结果包括了已有广义Baouendi-Grushin向量场的Hardy不等式．作为应用，讨论了由Baouendi-GrusMn向量场构成一退化次椭圆算子的一些性质和刻画了这类向量场构成的非线性算子的一个正解．     

Burkholder-Gundy-Davis inequality in martingale Hardy spaces with variable exponent

In this article, by extending classical Dellacherie's theorem on stochastic sequences to variable exponent spaces, we prove that the famous Burkholder-Gundy-Davis inequality holds for martingales in variable exponent Hardy spaces. We also obtain the variable exponent analogues of several martingale inequalities in classical theory, including convexity lemma, Chevalier's inequality and the equivalence of two kinds of martingale spaces with predictable control. Moreover, under the regular condition on σ-algebra sequence we prove the equivalence between five kinds of variable exponent martingale Hardy spaces.     

H型群内上半空间Hardy不等式的最佳常数问题

本文研究了H型幂零李群上Hardy不等式的问题.利用基本解的方法,获得了相关李群上的Hardy不等式,并且所得到的相关Hardy常数是最佳的.     

Estimates of Oscillations of the Hardy Transform

For the Hardy transform of a nonincreasing function we obtain a sharp two-sided estimate of the BLO-norm and sharp inequalities between the BMO- and the BLO-norms of a nonincreasing function. A well-known lower bound for the BMO-norm of the Hardy transform is improved on the basis of these inequalities.     

On Hardy-Type Inequalities and Hankel Transforms

 For the real Hardy spaces , we shall show the Hardy type integral inequalities, and applying the inequalities we shall establish the Hardy’s inequalities with respect to Hankel transforms. Received 21 October 1997; in revised form 19 October 1998     

Sharp weighted Hardy type inequalities and Hardy–Sobolev type inequalities on polarizable Carnot groups

In this Note, we establish sharp weighted Hardy type inequalities with a more general index p on polarizable Carnot groups, which include Kombe's recent results; then a weighted Hardy–Sobolev type inequality is obtained by using previous inequalities. To cite this article: J. Wang, P. Niu, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Hardy-Sobolev type inequalities on the H-type group

Motivated by the idea of Badiale and Tarantello who have found Hardy-Sobolev inequalities on Rⁿ, a class of Hardy-Sobolev type inequalities on H-type groups is proved via a new representation formula for functions. Extremal functions realizing equality in the inequalities are discussed by refined Concentration-Compactness principles. Finally, some sharp constants for Hardy type inequalities are given. The project supported by National Natural Science Foundation of China, Grant No. 10371099.     

Further bounds for hardy type differences

In this paper we define a functional as a difference between the right-hand side and left-hand side of the refined Boas type inequality using the notation of superquadratic and subquadratic functions and study its properties, such as exponential and logarithmic convexity. We also, state and prove improvements and reverses of new weighted Boas type inequalities. As a special case of our result we obtain improvements and reverses of the Hardy inequality and its dual inequality. We introduce new Cauchy type mean and prove monotonicity property of this mean.     

Hardy–Littlewood–Sobolev and Stein–Weiss inequalities on homogeneous Lie groups

In this note, we prove the Stein–Weiss inequality on general homogeneous Lie groups. The obtained results extend previously known inequalities. Special properties of homogeneous norms play a key role in our proofs. Also, we give a simple proof of the Hardy–Littlewood–Sobolev inequality on general homogeneous Lie groups.     

Heisenberg型群上一类精确Hardy型不等式和Hardy-Sobolev型不等式及应用(英文)

本文在Heisenberg型群上建立了一类精确的Hardy型不等式。采用的技巧是逼近及正则化的方法。进一步利用这个结果,本文建立了一类精确的Hardy-Sobolev型不等式。这两个结果包括了已有的相关结果。作为应用,讨论了一类具有Hardy位势的非线性算子的正定性与下无界性。