Hardy Inequality Research Articles

Optimal Hardy Inequality for Fractional Laplacians on the Integers

We study the fractional Hardy inequality on the integers. We prove the optimality of the Hardy weight and hence affirmatively answer the question of sharpness of the constant.

A note on the fractional Hardy inequality

We give a direct proof of fractional Hardy inequality by means of Littlewood–Paley decomposition and properties of singular homogeneous kernels of degree -d. A refinement when q>2 is proved.

On the improvements of Hardy and Copson inequalities

On the improvements of Hardy and Copson inequalities

Weighted weak-type iterated and bilinear Hardy inequalities

We characterize the good weights for a certain weighted weak-type iterated Hardy inequality to hold. As a consequence, we get the characterizations of some weighted weak-type bilinear Hardy inequalities.

Hardy-Leindler Type Inequalities for Multiple Integrals on Time Scales

Abstract Hardy-Leindler type inequalities and their converses for multiple integrals on time scales are proved by using Fubini’s theorem and induction principle. Some generalized versions of Hardy, Wirtinger and Leindler inequalities in both continuous and discrete cases are also derived in seek of applications.

Horizontal magnetic fields and improved Hardy inequalities in the Heisenberg group

In this article, we introduce a notion of magnetic field in the Heisenberg group and we study its influence on spectral properties of the corresponding magnetic (sub-elliptic) Laplacian. We show that uniform magnetic fields uplift the bottom of the spectrum. For magnetic fields vanishing at infinity, including Aharonov–Bohm potentials, we derive magnetic improvements to a variety of Hardy-type inequalities for the Heisenberg sub-Laplacian. In particular, we establish a sub-Riemannian analogue of Laptev and Weidl sub-criticality result for magnetic Laplacians in the plane. Instrumental for our argument is the validity of a Hardy-type inequality for the Folland–Stein operator, that we prove in this article and has an interest on its own.

Multipolar Hardy inequalities and mutual interaction of the poles

AbstractIn this paper we state the weighted Hardy inequality $$\begin{aligned} c\int _{{{\mathbb {R}}}^N}\sum _{i=1}^n \frac{\varphi ^2 }{|x-a_i|^2}\, \mu (x)dx\le \int _{{{\mathbb {R}}}^N} |\nabla \varphi |^2 \, \mu (x)dx +k \int _{\mathbb {R}^N}\varphi ^2 \, \mu (x)dx \end{aligned}$$ c ∫ R N ∑ i = 1 n φ 2 | x - a i | 2 μ ( x ) d x ≤ ∫ R N | ∇ φ | 2 μ ( x ) d x + k ∫ R N φ 2 μ ( x ) d x for any $$ \varphi $$ φ in a weighted Sobolev spaces, with $$c\in ]0,c_o[$$ c ∈ ] 0 , c o [ where $$c_o=c_o(N,\mu )$$ c o = c o ( N , μ ) is the optimal constant, $$a_1,\ldots ,a_n\in \mathbb {R}^N$$ a 1 , … , a n ∈ R N , k is a constant depending on $$\mu $$ μ . We show the relation between c and the closeness to the single pole. To this aim we analyze in detail the difficulties to be overcome to get the inequality.

On the fractional Korn inequality in bounded domains: Counterexamples to the case ps < 1

Abstract The validity of Korn’s first inequality in the fractional setting in bounded domains has been open. We resolve this problem by proving that in fact Korn’s first inequality holds in the case p s > 1 ps\gt 1 for fractional W 0 s , p ( Ω ) {W}_{0}^{s,p}\left(\Omega ) Sobolev fields in open and bounded C 1 {C}^{1} -regular domains Ω ⊂ R n \Omega \subset {{\mathbb{R}}}^{n} . Also, in the case p s < 1 ps\lt 1 , for any open bounded C 1 {C}^{1} domain Ω ⊂ R n \Omega \subset {{\mathbb{R}}}^{n} , we construct counterexamples to the inequality, i.e., Korn’s first inequality fails to hold in bounded domains. The proof of the inequality in the case p s > 1 ps\gt 1 follows a standard compactness approach adopted in the classical case, combined with a Hardy inequality, and a recently proven Korn second inequality by Mengesha and Scott [A Fractional Korn-type inequality for smooth domains and a regularity estimate for nonlinear nonlocal systems of equations, Commun. Math. Sci. 20 (2022), no. 2, 405–423]. The counterexamples constructed in the case p s < 1 ps\lt 1 are interpolations of a constant affine rigid motion inside the domain away from the boundary and of the zero field close to the boundary.

Improved Hardy Inequalities with a Class of Weights

In the framework of Hardy type inequalities and their applications to evolution problems, the paper deals with local and nonlocal weighted improved Hardy inequalities related to the study of Kolmogorov operators perturbed by singular potentials. The class of weights is wide enough. We focus our attention on weighted Hardy inequalities with potentials obtained by inverse square potentials adding a nonnegative correction term. The method used to get the results is based on the introduction of a suitable vector-valued function and on a generalized vector field method. The local estimates show some examples of this type of potentials and extend some known results to the weighted case.

Criticality of Schrödinger forms and recurrence of Dirichlet forms

Introducing the notion of extended Schrödinger spaces, we define the criticality and subcriticality of Schrödinger forms in the manner similar to the recurrence and transience of Dirichlet forms. We show that a Schrödinger form is critical (resp. subcritical) if and only if there exists an excessive function of the associated Schrödinger semigroup and the Dirichlet form defined by h h -transform of the excessive function is recurrent (resp. transient). We give an analytical condition for the subcriticality of Schrödinger forms in terms of the bottom of spectrum. We introduce a subclass K H {\mathcal {K}}_H of the local Kato class and show a Schrödinger form with potential in K H {\mathcal {K}}_H is critical. Critical Schrödinger forms lead us to critical Hardy-type inequalities. As an example, we treat fractional Schrödinger operators with potential in K H {\mathcal {K}}_H and reconsider the classical Hardy inequality by our approach.

Failure of the local chain rule for the fractional variation

We prove that the local version of the chain rule cannot hold for the fractional variation defined in in our previous article (2019). In the case n = 1 , we prove a stronger result, exhibiting a function f \in BV^{\alpha}(\mathbb{R}) such that \vert f \vert \notin BV^{\alpha}(\mathbb{R}) . The failure of the local chain rule is a consequence of some surprising rigidity properties for non-negative functions with bounded fractional variation which, in turn, are derived from a fractional Hardy inequality localized to half-spaces. Our approach exploits the distributional techniques developed in our previous works (2019–2022). As a byproduct, we refine the fractional Hardy inequality obtained in works of Shieh and Spector (2018) and Spector (2020) and we prove a fractional version of the closely related Meyers–Ziemer trace inequality.

Hardy Inequality in Variable Grand Lebesgue Spaces for Nonincreasing Functions

Hardy Inequality in Variable Grand Lebesgue Spaces for Nonincreasing Functions

A benchmark-type generalization of the Sugeno integral with applications in bibliometrics

We propose a new generalization of the classical Sugeno integral motivated by the Hirsch, Woeginger, and other geometrically-inspired indices of scientific impact. The new integral adapts to the rank-size curve better as it allows for putting more emphasis on highly-valued items and/or the tail of the distribution (level measure). We study its fundamental properties and give the conditions guaranteeing the fulfillment of subadditivity as well as the Jensen, Liapunov, Hardy, Markov, and Paley-Zygmund type inequalities. We discuss its applications in scientometrics.

Remarks on Some Higher Dimensional Hardy Inequalities

In this note, we give an elementary proof of Hardy inequality in higher dimensions introduced by Christ and Grafakos. The advantage of our approach is that it uses the one-dimensional Hardy inequality to obtain higher dimensional versions. We go further and get some well-known weighted estimates using the same approach.

Hardy and Rellich inequality on lattices

In this paper, we study the asymptotic behaviour of the sharp constant in discrete Hardy and Rellich inequality on the lattice mathbb {Z}^d as d rightarrow infty . In the process, we proved some Hardy-type inequalities for the operators Delta ^m and nabla (Delta ^m) for non-negative integers m on a d dimensional torus. It turns out that the sharp constant in discrete Hardy and Rellich inequality grows as d and d^2 respectively as d rightarrow infty .

Existence of triple solutions for elliptic equations driven by p-Laplacian-like operators with Hardy potential under Dirichlet–Neumann boundary conditions

In this article, we focus on triple weak solutions for some p-Laplacian-type elliptic equations with Hardy potential, two parameters, and mixed boundary conditions. We show the existence of at least three distinct weak solutions by using variational methods, the Hardy inequality, and the Bonanno–Marano-type three critical points theorem under suitable assumptions, and the existence of solutions to some particular cases of this type of elliptic equations are also obtained.

Weighted Hardy and Rellich Types Inequalities on the Heisenberg Group with Sharp Constants

This paper aims at deriving some weighted Hardy type and Rellich type inequalities with sharp constants on the Heisenberg group. The improved versions of these inequalities are established as well. The technique adopted involve the application of some elementary vectorial inequalities and some properties of Heisenberg group.

A remark on Davies' Hardy inequality

Мы приводим доказательство методом «интегрированием по частям» неравенства Харди в версии Дэвиса. При этом получается усиление со значительно большим весом Харди.

Some new hardy inequalities in probability

Some new hardy inequalities in probability

Hardy Inequality in Variable Grand Lebesgue Spaces for Nonincreasing Functions

Исследуется ограниченность оператора усреднения Харди между взвешенными гранд-пространствами Лебега с переменным показателем для невозрастающих функций. Библиография: 26 названий.