INEQUALITIES ASSOCIATED WITH DILATIONS

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).

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BEST CONSTANTS AND POHOZAEV IDENTITY FOR HARDY–SOBOLEV-TYPE OPERATORS

Communications in Contemporary Mathematics ◽

10.1142/s0219199712500502 ◽

2013 ◽

Vol 15 (03) ◽

pp. 1250050 ◽

Cited By ~ 4

Author(s):

ADIMURTHI

Keyword(s):

Hardy Inequality ◽

Direct Proof ◽

Sobolev Type ◽

General Elliptic ◽

Functional Aspects ◽

Approximation Lemma ◽

Definition Of ◽

Multiparticle Systems ◽

The Hardy Inequality ◽

Optimal Inequalities

This paper is threefold. Firstly, we reformulate the definition of the norm induced by the Hardy inequality (see [J. L. Vázquez and N. B. Zographopoulos, Functional aspects of the Hardy inequality. Appearance of a hidden energy, preprint (2011); http://arxiv. org/abs/1102.5661]) to more general elliptic and sub-elliptic Hardy–Sobolev-type operators. Secondly, we derive optimal inequalities (see [C. Cowan, Optimal inequalities for general elliptic operator with improvement, Commun. Pure Appl. Anal.9(1) (2010) 109–140; N. Ghoussoub and A. Moradifam, Bessel pairs and optimal Hardy and Hardy–Rellich inequalities, Math. Ann.349(1) (2010) 1–57 (electronic)]) for multiparticle systems in ℝN and Heisenberg group. In particular, we provide a direct proof of an optimal inequality with multipolar singularities shown in [R. Bossi, J. Dolbeault and M. J. Esteban, Estimates for the optimal constants in multipolar Hardy inequalities for Schrödinger and Dirac operators, Commun. Pure Appl. Anal.7(3) (2008) 533–562]. Finally, we prove an approximation lemma which allows to show that the domain of the Dirichlet–Laplace operator is dense in the domain of the corresponding Hardy operators. As a consequence, in some particular cases, we justify the Pohozaev-type identity for such operators.

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The Hardy inequality and second order nonlinear eigenvalue problems

Mathematical Surveys and Monographs - Functional Inequalities: New Perspectives and New Applications ◽

10.1090/surv/187/05 ◽

2013 ◽

pp. 59-68

Author(s):

Nassif Ghoussoub ◽

Amir Moradifam

Keyword(s):

Hardy Inequality ◽

Eigenvalue Problems ◽

Second Order ◽

Nonlinear Eigenvalue Problems ◽

The Hardy Inequality ◽

Nonlinear Eigenvalue

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

Communications in Contemporary Mathematics ◽

10.1142/s0219199720500248 ◽

2020 ◽

pp. 2050024 ◽

Cited By ~ 1

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.

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