scholarly journals Three weights higher order Hardy type inequalities

2006 ◽  
Vol 4 (2) ◽  
pp. 163-191
Author(s):  
Aigerim A. Kalybay ◽  
Lars-Erik Persson
Keyword(s):  
Differential Operator ◽  
Weight Function ◽  
Type Inequality ◽  
Higher Order ◽  
Hardy Type Inequality ◽  
Volterra Type ◽  
Crucial Step ◽  
Hardy Type Inequalities ◽  
Hardy Type ◽  
Volterra Type Operator

We investigate the following three weights higher order Hardy type inequality (0.1)‖g‖q,u≤  C‖Dρkg‖p,vwhereDρidenotes the following weighted differential operator:{dig(t)dti,i=0,1,...,m−1,di−mdti−m(p(t)dmg(t)dtm),i=m,m+1,...,k,for a weight functionρ(⋅). A complete description of the weightsu,vandρso that (0.1) holds was given in [4] for the case1<p≤q<∞. Here the corresponding characterization is proved for the case1<q<p<∞. The crucial step in the proof of the main result is to use a new Hardy type inequality (for a Volterra type operator), which we first state and prove.

scholarly journals Characterizations of weighted dynamic Hardy-type inequalities with higher-order derivatives

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
S. H. Saker ◽  
R. R. Mahmoud ◽  
K. R. Abdo
Keyword(s):  
Time Scales ◽  
Sufficient Conditions ◽  
Type Inequality ◽  
Higher Order ◽  
Necessary And Sufficient Conditions ◽  
Hardy Type Inequality ◽  
Hardy Type Inequalities ◽  
Hardy Type ◽  
Weighted Functions ◽  
Necessary And Sufficient

AbstractIn this paper, we establish some necessary and sufficient conditions for the validity of a generalized dynamic Hardy-type inequality with higher-order derivatives with two different weighted functions on time scales. The corresponding continuous and discrete cases are captured when $\mathbb{T=R}$ T = R and $\mathbb{T=N}$ T = N , respectively. Finally, some applications to our main result are added to conclude some continuous results known in the literature and some other discrete results which are essentially new.

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

2010 ◽  
Vol 62 (5) ◽  
pp. 1116-1130 ◽  
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.

scholarly journals A Higher-Order Hardy-Type Inequality in Anisotropic Sobolev Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-7 ◽  
Author(s):  
Paolo Secchi
Keyword(s):  
Sobolev Spaces ◽  
Hyperbolic Systems ◽  
Type Inequality ◽  
Initial Boundary ◽  
Higher Order ◽  
Hardy Type Inequality ◽  
Characteristic Boundary ◽  
Hardy Type ◽  
Anisotropic Sobolev Spaces ◽  
Initial Boundary Value

We prove a higher-order inequality of Hardy type for functions in anisotropic Sobolev spaces that vanish at the boundary of the space domain. This is an important calculus tool for the study of initial-boundary-value problems of symmetric hyperbolic systems with characteristic boundary.

A Sharp Adams-Type Inequality for Weighted Sobolev Spaces

2020 ◽  
Vol 71 (2) ◽  
pp. 517-538
Author(s):  
João Marcos do Ó ◽  
Abiel Costa Macedo ◽  
José Francisco de Oliveira
Keyword(s):  
Sobolev Spaces ◽  
Weighted Sobolev Spaces ◽  
Type Inequality ◽  
Higher Order ◽  
Sobolev Type ◽  
Hardy Type Inequality ◽  
Hardy Type ◽  
Classical Work

Abstract In a classical work (Ann. Math.128, (1988) 385–398), D. R. Adams proved a sharp Trudinger–Moser inequality for higher-order derivatives. We derive a sharp Adams-type inequality and Sobolev-type inequalities associated with a class of weighted Sobolev spaces that is related to a Hardy-type inequality.

scholarly journals A class of weighted Hardy type inequalities in $$\mathbb {R}^N$$

2021 ◽  
Author(s):  
Anna Canale
Keyword(s):  
Weighted Sobolev Space ◽  
Type Inequality ◽  
Field Method ◽  
Weight Functions ◽  
Singular Potentials ◽  
Hardy Type Inequality ◽  
Hardy Type Inequalities ◽  
Hardy Type ◽  
Inverse Square Potentials ◽  
Vector Field Method

AbstractIn the paper we prove the weighted Hardy type inequality $$\begin{aligned} \int _{{{\mathbb {R}}}^N}V\varphi ^2 \mu (x)dx\le \int _{\mathbb {R}^N}|\nabla \varphi |^2\mu (x)dx +K\int _{\mathbb {R}^N}\varphi ^2\mu (x)dx, \end{aligned}$$ ∫ R N V φ 2 μ ( x ) d x ≤ ∫ R N | ∇ φ | 2 μ ( x ) d x + K ∫ R N φ 2 μ ( x ) d x , for functions $$\varphi $$ φ in a weighted Sobolev space $$H^1_\mu $$ H μ 1 , for a wider class of potentials V than inverse square potentials and for weight functions $$\mu $$ μ of a quite general type. The case $$\mu =1$$ μ = 1 is included. To get the result we introduce a generalized vector field method. The estimates apply to evolution problems with Kolmogorov operators $$\begin{aligned} Lu=\varDelta u+\frac{\nabla \mu }{\mu }\cdot \nabla u \end{aligned}$$ L u = Δ u + ∇ μ μ · ∇ u perturbed by singular potentials.

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.

scholarly journals Some New Iterated Hardy-Type Inequalities

2012 ◽  
Vol 2012 ◽  
pp. 1-30 ◽  
Author(s):  
A. Gogatishvili ◽  
R. CH. Mustafayev ◽  
L.-E. Persson
Keyword(s):  
Type Inequality ◽  
Independent Interest ◽  
Weight Functions ◽  
Hardy Type Inequality ◽  
Hardy Type Inequalities ◽  
Hardy Type

We characterize the validity of the Hardy-type inequality∥∥∫s∞h(z)dz   ∥p,u,(0,t)   ∥q,w,(0,  ∞)≤c  ∥h∥θ,v(0,∞), where0<p<∞,0<q≤∞,1<θ≤∞,u,w, andvare weight functions on(0,∞). Some fairly new discretizing and antidiscretizing techniques of independent interest are used.

scholarly journals Pseudo-monotonicity and degenerated or singular elliptic operators

1998 ◽  
Vol 58 (2) ◽  
pp. 213-221 ◽  
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.

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

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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