Hardy type inequalities for the fractional relativistic operator

<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>

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

Mathematical Problems in Engineering ◽

10.1155/2021/9939468 ◽

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.

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An Extension Problem and Hardy Type Inequalities for the Grushin Operator

10.1007/978-3-030-72058-2_1 ◽

2021 ◽

pp. 1-28

Author(s):

Rakesh Balhara ◽

Pradeep Boggarapu ◽

Sundaram Thangavelu

Keyword(s):

Extension Problem ◽

Hardy Type Inequalities ◽

Grushin Operator ◽

Hardy Type

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Anisotropic Hardy inequalities

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210517000336 ◽

2017 ◽

Vol 148 (3) ◽

pp. 483-498 ◽

Cited By ~ 1

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.

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Hardy Inequalities on the Real Line

Canadian Mathematical Bulletin ◽

10.4153/cmb-2010-091-8 ◽

2011 ◽

Vol 54 (1) ◽

pp. 159-171 ◽

Cited By ~ 1

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.

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Reverses Hardy-type inequalities via Jensen integral inequality

Mathematica Montisnigri ◽

10.20948/mathmontis-2021-52-5 ◽

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.

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

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/prm.2019.37 ◽

2019 ◽

Vol 150 (6) ◽

pp. 2952-2981 ◽

Cited By ~ 2

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.

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Hardy-Type Inequalities for Fractional Powers of the Dunkl–Hermite Operator

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091517000311 ◽

2018 ◽

Vol 61 (2) ◽

pp. 513-544 ◽

Cited By ~ 3

Author(s):

Óscar Ciaurri ◽

Luz Roncal ◽

Sundaram Thangavelu

Keyword(s):

Spectral Theorem ◽

Fractional Powers ◽

Hermite Operator ◽

Hardy Type Inequalities ◽

Hardy Type ◽

Push Forward ◽

Non Local ◽

Euclidean Setting ◽

Fractional Harmonic Oscillator ◽

Fractional Type

AbstractWe prove Hardy-type inequalities for a fractional Dunkl–Hermite operator, which incidentally gives Hardy inequalities for the fractional harmonic oscillator as well. The idea is to useh-harmonic expansions to reduce the problem in the Dunkl–Hermite context to the Laguerre setting. Then, we push forward a technique based on a non-local ground representation, initially developed by Franket al. [‘Hardy–Lieb–Thirring inequalities for fractional Schrödinger operators,J. Amer. Math. Soc.21(2008), 925–950’] in the Euclidean setting, to obtain a Hardy inequality for the fractional-type Laguerre operator. The above-mentioned method is shown to be adaptable to an abstract setting, whenever there is a ‘good’ spectral theorem and an integral representation for the fractional operators involved.

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