scholarly journals Improved Hardy-Rellich inequalities

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Biagio Cassano ◽  
Lucrezia Cossetti ◽  
Luca Fanelli
Keyword(s):  
Magnetic Field ◽  
Hardy Inequality ◽  
Best Constant ◽  
Rellich Inequality ◽  
Free Case ◽  
Optimal Inequality ◽  
Free Operator ◽  
The Hardy Inequality

<p style='text-indent:20px;'>We investigate Hardy-Rellich inequalities for perturbed Laplacians. In particular, we show that a non-trivial angular perturbation of the free operator typically improves the inequality, and may also provide an estimate which does not hold in the free case. The main examples are related to the introduction of a magnetic field: this is a manifestation of the diamagnetic phenomenon, which has been observed by Laptev and Weidl in [<xref ref-type="bibr" rid="b21">21</xref>] for the Hardy inequality, later by Evans and Lewis in [<xref ref-type="bibr" rid="b9">9</xref>] for the Rellich inequality; however, to the best of our knowledge, the so called Hardy-Rellich inequality has not yet been investigated in this regards. After showing the optimal inequality, we prove that the best constant is not attained by any function in the domain of the estimate.</p>

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 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—Poincaré inequalities with boundary singularities

2012 ◽  
Vol 142 (4) ◽  
pp. 769-786 ◽  
Author(s):  
Mouhamed Moustapha Fall ◽  
Roberta Musina
Keyword(s):  
Variational Problem ◽  
Bounded Domain ◽  
Hardy Inequality ◽  
Necessary Conditions ◽  
Poincaré Inequality ◽  
Variational Problems ◽  
Poincare Inequality ◽  
Best Constant ◽  
Sufficient And Necessary Conditions ◽  
The Hardy Inequality

We are interested in variational problems involving weights that are singular at a point of the boundary of the domain. More precisely, we study a linear variational problem related to the Poincaré inequality and to the Hardy inequality for maps in H01(Ω), where Ω is a bounded domain in ℝN, N ≥ 2, with 0 ∈ ∂Ω. In particular, we give sufficient and necessary conditions so that the best constant is achieved.

scholarly journals Fractional Hardy–Sobolev Inequalities with Magnetic Fields

2019 ◽  
Vol 2019 ◽  
pp. 1-5
Author(s):  
Min Liu ◽  
Fengli Jiang ◽  
Zhenyu Guo
Keyword(s):  
Magnetic Field ◽  
Magnetic Fields ◽  
Sobolev Inequality ◽  
Sobolev Inequalities ◽  
Best Constant

A fractional Hardy–Sobolev inequality with a magnetic field is studied in the present paper. Under appropriate conditions, the achievement of the best constant of the fractional magnetic Hardy–Sobolev inequality is established.

scholarly journals The Prehistory of the Hardy Inequality

2006 ◽  
Vol 113 (8) ◽  
pp. 715 ◽  
Author(s):  
Alois Kufner ◽  
Lech Maligranda ◽  
Lars-Erik Persson
Keyword(s):  
Hardy Inequality ◽  
The Hardy Inequality

scholarly journals INEQUALITIES ASSOCIATED WITH DILATIONS

2009 ◽  
Vol 11 (02) ◽  
pp. 265-277 ◽  
Author(s):  
TOHRU OZAWA ◽  
HIRONOBU SASAKI
Keyword(s):  
Hardy Inequality ◽  
Sobolev Type ◽  
Semi Group ◽  
The Hardy Inequality

Some properties of distributions f satisfying x · ∇ f ∈ Lp (ℝn), 1 ≤ p < ∞, are studied. The operator x · ∇ is the generator of a semi-group of dilations. We first give Sobolev type inequalities with respect to the operator x · ∇. Using the inequalities, we also show that if [Formula: see text], x · ∇ f ∈ Lp (ℝn) and |x|n/p|f(x)| vanishes at infinity, then f belongs to Lp (ℝn). One of the Sobolev type inequalities is shown to be equivalent to the Hardy inequality in L2 (ℝn).

The Hardy Inequality for Hermite Expansions

2014 ◽  
Vol 21 (2) ◽  
pp. 267-280 ◽  
Author(s):  
Zhongkai Li ◽  
Yufeng Yu ◽  
Yehao Shi
Keyword(s):  
Hardy Inequality ◽  
Hermite Expansions ◽  
The Hardy Inequality
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