Hardy's Inequalities for Hermite and Laguerre Expansions

The well-known inequality of Hardy for Fourier coefficientsof functions in the real Hardy space is . We shall establish analogues of this inequality for the Hermite function expansionsand also for the Laguerre function expansions. 1991 MathematicsSubject Classification 42C10, 33C45.     

Hardy’s Inequality for Laguerre Expansions of Hermite Type

Journal of Fourier Analysis and Applications - Hardy’s inequality for Laguerre expansions of Hermite type with the index $$\alpha \in (\{-1/2\}\cup [1/2,\infty ))^d$$ is proved in the...     

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。     

The equivalence between pointwise Hardy inequalities and uniform fatness

We prove an equivalence result between the validity of a pointwise Hardy inequality in a domain and uniform capacity density of the complement. This result is new even in Euclidean spaces, but our methods apply in general metric spaces as well. We also present a new transparent proof for the fact that uniform capacity density implies the classical integral version of the Hardy inequality in the setting of metric spaces. In addition, we consider the relations between the above concepts and certain Hausdorff content conditions.     

关于一类向量场的Picone恒等式和Hardy不等式1

本文给出了构成Baouendi-Grushin算子 的向量场的Picone恒 式,由此导出了Hardy不等式.已有文献中得到的不等式成为本文结果的特殊情形。     

Perturbation of the CR fractional Yamabe problem

In this note, we first prove the non‐degeneracy property of extremals for the optimal Hardy–Littlewood–Sobolev inequality on the Heisenberg group, as an application, a perturbation result for the CR fractional Yamabe problem is obtained, this generalizes a classical result of Malchiodi and Uguzzoni 30 .     

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.     

Riesz-Fejér Inequalities for Harmonic Functions

Potential Analysis - In this article, we prove the Riesz - Fejér inequality for complex-valued harmonic functions in the harmonic Hardy space hp for all p >?1. The result is sharp...     

Hardy平均及其不等式

本文研究了涉及Hardy平均的不等式问题.借助于优超理论,获得了2维Hardy平均不等式成立的充要条件及n维Hardy平均不等式的充分条件.并为建立解析不等式和高维结果提供了一些方法.     

An optimal Hardy–Morrey inequality

In this work we improve the sharp Hardy inequality in the case p?>?n by adding an optimal weighted H?lder seminorm. To achieve this we first obtain a local improvement. We also obtain a refinement of both the Sobolev inequality for p?>?n and the Hardy inequality, the latter having the best constant.     

A Hardy type inequality in the half-space on and Heisenberg group

We prove a Hardy type inequality in the half-space on the Heisenberg group and show that a Hardy inequality given by J. Tidblm in [J. Tidblm, A Hardy inequality in the half-space, J. Funct. Anal. 221 (2005) 482–492] is sharp.     

Hardy inequalities for fractional integrals on general domains

We prove a sharp Hardy inequality for fractional integrals for functions that are supported in a general domain. The constant is the same as the one for the half-space and hence our result settles a recent conjecture of Bogdan and Dyda.     

Hardy–Sobolev Type Inequalities with Sharp Constants in Carnot–Carathéodory Spaces

We prove a generalization with sharp constants of a classical inequality due to Hardy to Carnot groups of arbitrary step, or more general Carnot–Carathéodory spaces associated with a system of vector fields of Hörmander type. Under a suitable additional assumption (see Eq. 1.6 below) we are able to extend such result to the nonlinear case \(p\not= 2\). We also obtain a sharp inequality of Hardy–Sobolev type.     

Existence results for elliptic problems with Hardy potential

In this work we obtain existence results for some singular elliptic Dirichlet problems involving the p-Laplacian. Precisely, starting from a weak lower semicontinuity result and by using the classical Hardy inequality, a critical point result for differentiable functionals is exploited, in order to prove the existence of a precise open interval of positive eigenvalues for which the treated problems admit at least one non-trivial weak solution.     

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．     

Fourier Restriction of H<Superscript>1</Superscript> Functions on Polynomial Surfaces

We prove a Fourier restriction theorem for H¹ functions, which is a generalization of a classical inequality of Hardy. As a consequence of our result, we explicitly obtain a class of BMO functions.     

Hardy's inequalities for Hermite and Laguerre expansions

Hardy's inequalities are proved for higher-dimensional Hermite and special Hermite expansions of functions in Hardy spaces. Inequalities for multiple Laguerre expansions are also deduced.

     

Hardy's inequality and the boundary size

We establish a self-improving property of the Hardy inequality and an estimate on the size of the boundary of a domain supporting a Hardy inequality.

     

On Hardy inequalities with a remainder term

In this paper we study some improvements of the classical Hardy inequality. We add to the right hand side of the inequality a term that depends on some Lorentz norms of u or of its gradient and we find the best values of the constants for remaining terms. In both cases we show that the problem of finding the optimal value of the constant can be reduced to a spherically symmetric situation. This result is new when the right hand side is a Lorentz norm of the gradient.     

Two new forms of half-discrete Hilbert inequality

In this paper, we introduce two new forms of the half-discrete Hilbert inequality. The first form is a sharper form of the half-discrete Hilbert inequality and is related to Hardy inequality. In the second one, we give a differential form of this inequality.