A sharp integral Hardy type inequality and applications to Muckenhoupt weights on R

Abstract In this paper, we will provide a complete study of the self-improving properties of the discrete Muckenhoupt class 𝒜 p ⁢ ( 𝒞 ) {\mathcal{A}^{p}(\mathcal{C})} of weights defined on ℤ + {\mathbb{Z}_{+}} . In addition, we will determine the range of the new constants which are related to the original constants via an algebraic equation. For illustration, we will give an example to prove that the results are sharp. The results will be obtained by employing a discrete version of an inequality due to Hardy–Littlewood and a new discrete Hardy-type inequality with negative powers.

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Pseudo-monotonicity and degenerated or singular elliptic operators

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700032184 ◽

1998 ◽

Vol 58 (2) ◽

pp. 213-221 ◽

Cited By ~ 9

Author(s):

P. Drábek ◽

A. Kufner ◽

V. Mustonen

Keyword(s):

Sobolev Spaces ◽

Differential Operators ◽

Weighted Sobolev Spaces ◽

Type Inequality ◽

Elliptic Operators ◽

Hardy Type Inequality ◽

Hardy Type ◽

Elliptic Differential Operators

Using the compactness of an imbedding for weighted Sobolev spaces (that is, a Hardy-type inequality), it is shown how the assumption of monotonicity can be weakened still guaranteeing the pseudo-monotonicity of certain nonlinear degenerated or singular elliptic differential operators. The result extends analogous assertions for elliptic operators.

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Eigencurves of the p(·)-Biharmonic operator with a Hardy-type term

Moroccan Journal of Pure and Applied Analysis ◽

10.2478/mjpaa-2020-0015 ◽

2020 ◽

Vol 6 (2) ◽

pp. 198-209

Author(s):

Mohamed Laghzal ◽

Abdelouahed El Khalil ◽

My Driss Morchid Alaoui ◽

Abdelfattah Touzani

Keyword(s):

Dirichlet Problem ◽

Variational Approach ◽

Type Inequality ◽

Biharmonic Operator ◽

Nondecreasing Sequence ◽

Hardy Type Inequality ◽

Principal Curve ◽

Hardy Type ◽

Homogeneous Dirichlet Problem

AbstractThis paper is devoted to the study of the homogeneous Dirichlet problem for a singular nonlinear equation which involves the p(·)-biharmonic operator and a Hardy-type term that depend on the solution and with a parameter λ. By using a variational approach and min-max argument based on Ljusternik-Schnirelmann theory on C1-manifolds [13], we prove that the considered problem admits at least one nondecreasing sequence of positive eigencurves with a characterization of the principal curve μ1(λ) and also show that, the smallest curve μ1(λ) is positive for all 0 ≤ λ < CH, with CH is the optimal constant of Hardy type inequality.

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DISCRETE HARDY-TYPE INEQUALITY WITH KERNEL

International Journal of Pure and Apllied Mathematics ◽

10.12732/ijpam.v106i2.11 ◽

2016 ◽

Vol 106 (2) ◽

Author(s):

S. Kumar

Keyword(s):

Type Inequality ◽

Hardy Type Inequality ◽

Hardy Type

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Corrigendum to "Hardy type inequality in variable Lebesgue spaces"

Annales Academiae Scientiarum Fennicae Mathematica ◽

10.5186/aasfm.2010.3542 ◽

2010 ◽

Vol 35 ◽

pp. 679-680 ◽

Cited By ~ 3

Author(s):

Humberto Rafeiro ◽

Stefan Samko

Keyword(s):

Type Inequality ◽

Lebesgue Spaces ◽

Hardy Type Inequality ◽

Variable Lebesgue Spaces ◽

Hardy Type

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Degenerate p-Laplacian Operators and Hardy Type Inequalities on H-Type Groups

Canadian Journal of Mathematics ◽

10.4153/cjm-2010-033-9 ◽

2010 ◽

Vol 62 (5) ◽

pp. 1116-1130 ◽

Cited By ~ 2

Author(s):

Yongyang Jin ◽

Genkai Zhang

Keyword(s):

Vector Fields ◽

Orthonormal Basis ◽

Type Inequality ◽

Laplacian Operator ◽

Hardy Type Inequality ◽

Invariant Vector ◽

Hardy Type Inequalities ◽

Hardy Type ◽

Left Invariant Vector ◽

Laplacian Operators

AbstractLet 𝔾 be a step-two nilpotent group of H-type with Lie algebra 𝔊 = V ⊕ t. We define a class of vector fields X = {Xj} on 𝔾 depending on a real parameter k ≥ 1, and we consider the corresponding p-Laplacian operator Lp,ku = divX(|∇Xu|p−2∇Xu). For k = 1 the vector fields X = {Xj} are the left invariant vector fields corresponding to an orthonormal basis of V; for 𝔾 being the Heisenberg group the vector fields are the Greiner fields. In this paper we obtain the fundamental solution for the operator Lp,k and as an application, we get a Hardy type inequality associated with X.

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