New sharp Hardy and Rellich type inequalities on Cartan–Hadamard manifolds and their improvements

2019 ◽  
Vol 150 (6) ◽  
pp. 2952-2981 ◽  
Author(s):  
Van Hoang Nguyen
Keyword(s):  
Sectional Curvature ◽  
Hyperbolic Space ◽  
Hardy Inequality ◽  
Hadamard Manifolds ◽  
Hardy Inequalities ◽  
Hardy Type Inequalities ◽  
Rellich Inequality ◽  
Hardy Type

In this paper, we prove several new Hardy type inequalities (such as the weighted Hardy inequality, weighted Rellich inequality, critical Hardy inequality and critical Rellich inequality) related to the radial derivation (i.e., the derivation along the geodesic curves) on the Cartan–Hadamard manifolds. By Gauss lemma, our new Hardy inequalities are stronger than the classical ones. We also establish the improvements of these inequalities in terms of sectional curvature of the underlying manifolds which illustrate the effect of curvature to these inequalities. Furthermore, we obtain some improvements of Hardy and Rellich inequalities on the hyperbolic space ℍn. Especially, we show that our new Rellich inequalities are indeed stronger than the classical ones on the hyperbolic space ℍn.

scholarly journals An optimal improvement for the Hardy inequality on the hyperbolic space and related manifolds

2019 ◽  
Vol 150 (4) ◽  
pp. 1699-1736 ◽  
Author(s):  
Elvise Berchio ◽  
Debdip Ganguly ◽  
Gabriele Grillo ◽  
Yehuda Pinchover
Keyword(s):  
Hyperbolic Space ◽  
Comparison Theorem ◽  
Uncertainty Principle ◽  
Hardy Inequality ◽  
Concentration Compactness ◽  
Hadamard Manifolds ◽  
Quantitative Version ◽  
Rellich Inequality ◽  
Hardy Type ◽  
The Hardy Inequality

AbstractWe prove optimal improvements of the Hardy inequality on the hyperbolic space. Here, optimal means that the resulting operator is critical in the sense of Devyver, Fraas, and Pinchover (2014), namely the associated inequality cannot be further improved. Such inequalities arise from more general, optimal ones valid for the operator $ P_{\lambda }:= -\Delta _{{\open H}^{N}} - \lambda $ where 0 ⩽ λ ⩽ λ1(ℍN) and λ1(ℍN) is the bottom of the L2 spectrum of $-\Delta _{{\open H}^{N}} $, a problem that had been studied in Berchio, Ganguly, and Grillo (2017) only for the operator $P_{\lambda _{1}({\open H}^{N})}$. A different, critical and new inequality on ℍN, locally of Hardy type is also shown. Such results have in fact greater generality since they are proved on general Cartan-Hadamard manifolds under curvature assumptions, possibly depending on the point. Existence/nonexistence of extremals for the related Hardy-Poincaré inequalities are also proved using concentration-compactness technique and a Liouville comparison theorem. As applications of our inequalities, we obtain an improved Rellich inequality and we derive a quantitative version of Heisenberg-Pauli-Weyl uncertainty principle for the operator $P_\lambda.$

scholarly journals Hardy-type inequalities for Dunkl operators with applications to many-particle Hardy inequalities

2020 ◽  
pp. 2050024 ◽  
Author(s):  
Andrei Velicu
Keyword(s):  
Hardy Inequality ◽  
Root Systems ◽  
Dunkl Operators ◽  
Hardy Inequalities ◽  
One Dimensional ◽  
Hardy Type Inequalities ◽  
Rellich Inequality ◽  
Nirenberg Inequality ◽  
Special Case ◽  
The Hardy Inequality

In this paper, we study various forms of the Hardy inequality for Dunkl operators, including the classical inequality, [Formula: see text] inequalities, an improved Hardy inequality, as well as the Rellich inequality and a special case of the Caffarelli–Kohn–Nirenberg inequality. As a consequence, one-dimensional many-particle Hardy inequalities for generalized root systems are proved, which in the particular case of root systems [Formula: see text] improve some well-known results.

scholarly journals Some Hardy-Type Inequalities for Superquadratic Functions via Delta Fractional Integrals

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Usama Hanif ◽  
Ammara Nosheen ◽  
Rabia Bibi ◽  
Khuram Ali Khan ◽  
Hamid Reza Moradi
Keyword(s):  
Fractional Calculus ◽  
Time Scales ◽  
Fractional Integrals ◽  
Discrete Fractional Calculus ◽  
Hardy Inequalities ◽  
Hardy Type Inequalities ◽  
Hardy Type ◽  
Superquadratic Functions

In this paper, Jensen and Hardy inequalities, including Pólya–Knopp type inequalities for superquadratic functions, are extended using Riemann–Liouville delta fractional integrals. Furthermore, some inequalities are proved by using special kernels. Particular cases of obtained inequalities give us the results on time scales calculus, fractional calculus, discrete fractional calculus, and quantum fractional calculus.

Anisotropic Hardy inequalities

2017 ◽  
Vol 148 (3) ◽  
pp. 483-498 ◽  
Author(s):  
Francesco Della Pietra ◽  
Giuseppina di Blasio ◽  
Nunzia Gavitone
Keyword(s):  
Distance Function ◽  
Hardy Inequalities ◽  
Hardy Type Inequalities ◽  
Hardy Type ◽  
General Norm

We study some Hardy-type inequalities involving a general norm in ℝn and an anisotropic distance function to the boundary. The case of the optimality of the constants is also addressed.

Hardy Inequalities on the Real Line

2011 ◽  
Vol 54 (1) ◽  
pp. 159-171 ◽  
Author(s):  
Mohammad Sababheh
Keyword(s):  
Real Line ◽  
Hardy’S Inequality ◽  
Hardy Inequalities ◽  
The Real ◽  
Hardy Type Inequalities ◽  
Hardy Type ◽  
Hardy's Inequality ◽  
Littlewood Conjecture ◽  
The Relationship

AbstractWe prove that some inequalities, which are considered to be generalizations of Hardy's inequality on the circle, can be modified and proved to be true for functions integrable on the real line. In fact we would like to show that some constructions that were used to prove the Littlewood conjecture can be used similarly to produce real Hardy-type inequalities. This discussion will lead to many questions concerning the relationship between Hardy-type inequalities on the circle and those on the real line.

scholarly journals Reverses Hardy-type inequalities via Jensen integral inequality

2021 ◽  
Vol 52 ◽  
pp. 43-51
Author(s):  
Bouharket Benaissa ◽  
Aissa Benguessoum
Keyword(s):  
Weight Function ◽  
Integral Inequality ◽  
Hölder Inequality ◽  
Integral Inequalities ◽  
Hardy Inequalities ◽  
Hardy Type Inequalities ◽  
Hardy Type ◽  
Jensen Integral Inequality ◽  
Number Of Authors

The integral inequalities concerning the inverse Hardy inequalities have been studied by a large number of authors during this century, of these articles have appeared, the work of Sulaiman in 2012, followed by Banyat Sroysang who gave an extension to these inequalities in 2013. In 2020 B. Benaissa presented a generalization of inverse Hardy inequalities. In this article, we establish a new generalization of these inequalities by introducing a weight function and a second parameter. The results will be proved using the Hölder inequality and the Jensen integral inequality. Several the reverses weighted Hardy’s type inequalities and the reverses Hardy’s type inequalities were derived from the main results.

scholarly journals Hardy type inequalities for the fractional relativistic operator

2022 ◽  
Vol 4 (3) ◽  
pp. 1-16
Author(s):  
Luz Roncal ◽  
◽  
◽  
Keyword(s):  
Ground State ◽  
Extension Problem ◽  
Local Version ◽  
State Representation ◽  
Hardy Inequalities ◽  
Hardy Type Inequalities ◽  
Hardy Type ◽  
Non Local

<abstract><p>We prove Hardy type inequalities for the fractional relativistic operator by using two different techniques. The first approach goes through trace Hardy inequalities. In order to get the latter, we study the solutions of the associated extension problem. The second develops a non-local version of the ground state representation in the spirit of Frank, Lieb, and Seiringer.</p></abstract>

Iterating Bilinear Hardy Inequalities

2017 ◽  
Vol 60 (4) ◽  
pp. 955-971 ◽  
Author(s):  
Martin Křepela
Keyword(s):  
Linear Operator ◽  
Simple Proof ◽  
Hardy Inequality ◽  
Integral Operators ◽  
Multilinear Operators ◽  
Hardy Inequalities ◽  
Hardy Type ◽  
Image Position ◽  
Iteration Technique

AbstractAn iteration technique for characterizing boundedness of certain types of multilinear operators is presented, reducing the problem to a corresponding linear-operator case. The method gives a simple proof of a characterization of validity of the weighted bilinear Hardy inequalityfor all non-negative f, g on (a, b), for 1 < p1, p2, q < ∞. More equivalent characterizing conditions are presented.The same technique is applied to various further problems, in particular those involving multilinear integral operators of Hardy type.

scholarly journals q-Hardy type inequalities for quantum integrals

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Necmettin Alp ◽  
Mehmet Zeki Sarikaya
Keyword(s):  
Hardy Inequality ◽  
Integral Inequality ◽  
Integral Inequalities ◽  
Hardy Type Inequalities ◽  
Quantum Calculus ◽  
Quantum Numbers ◽  
Hardy Type ◽  
Type Integral ◽  
The Hardy Inequality

AbstractThe aim of this work is to obtain quantum estimates for q-Hardy type integral inequalities on quantum calculus. For this, we establish new identities including quantum derivatives and quantum numbers. After that, we prove a generalized q-Minkowski integral inequality. Finally, with the help of the obtained equalities and the generalized q-Minkowski integral inequality, we obtain the results we want. The outcomes presented in this paper are q-extensions and q-generalizations of the comparable results in the literature on inequalities. Additionally, by taking the limit $q\rightarrow 1^{-}$ q → 1 − , our results give classical results on the Hardy inequality.

scholarly journals Fractional Korn and Hardy-type inequalities for vector fields in half space

2019 ◽  
Vol 21 (07) ◽  
pp. 1850055 ◽  
Author(s):  
Tadele Mengesha
Keyword(s):  
Half Space ◽  
Hardy Inequality ◽  
Vector Fields ◽  
A Priori ◽  
Type Inequality ◽  
Fractional Sobolev Spaces ◽  
Hardy Type Inequality ◽  
Hardy Type Inequalities ◽  
Korn’S Inequality ◽  
Hardy Type

We prove a fractional Hardy-type inequality for vector fields over the half space based on a modified fractional semi-norm. A priori, the modified semi-norm is not known to be equivalent to the standard fractional semi-norm and in fact gives a smaller norm, in general. As such, the inequality we prove improves the classical fractional Hardy inequality for vector fields. We will use the inequality to establish the equivalence of a space of functions (of interest) defined over the half space with the classical fractional Sobolev spaces, which amounts to prove a fractional version of the classical Korn’s inequality.

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