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.

Weighted norm inequalities for positive operators restricted on the cone of λ-quasiconcave functions

2019 ◽  
Vol 150 (1) ◽  
pp. 17-39 ◽  
Author(s):  
Amiran Gogatishvili ◽  
Júlio S. Neves
Keyword(s):  
Sufficient Conditions ◽  
Type Inequality ◽  
Weighted Norm ◽  
Hardy Type Inequality ◽  
Weighted Function ◽  
Hardy Type ◽  
Quasiconcave Functions ◽  
Measurable Functions ◽  
Image Position ◽  
Necessary And Sufficient

AbstractLet ρ be a monotone quasinorm defined on ${\rm {\frak M}}^ + $, the set of all non-negative measurable functions on [0, ∞). Let T be a monotone quasilinear operator on ${\rm {\frak M}}^ + $. We show that the following inequality restricted on the cone of λ-quasiconcave functions $$\rho (Tf) \les C_1\left( {\int_0^\infty {f^p} v} \right)^{1/p},$$where $1\les p\les \infty $ and v is a weighted function, is equivalent to slightly different inequalities considered for all non-negative measurable functions. The case 0 < p < 1 is also studied for quasinorms and operators with additional properties. These results in turn enable us to establish necessary and sufficient conditions on the weights (u, v, w) for which the three weighted Hardy-type inequality $$\left( {\int_0^\infty {{\left( {\int_0^x f u} \right)}^q} w(x){\rm d}x} \right)^{1/q} \les C_1\left( {\int_0^\infty {f^p} v} \right)^{1/p},$$holds for all λ-quasiconcave functions and all 0 < p, q ⩽ ∞.

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.

Partial Higher-Order Specifications1

1992 ◽  
Vol 16 (2) ◽  
pp. 101-126
Author(s):  
Egidio Astesiano ◽  
Maura Cerioli
Keyword(s):  
Sufficient Conditions ◽  
Higher Order ◽  
Necessary And Sufficient Conditions ◽  
Important Case ◽  
Deduction System ◽  
Closure Properties ◽  
Inference System ◽  
Conditional Specification ◽  
Necessary And Sufficient

In this paper the classes of extensional models of higher-order partial conditional specifications are studied, with the emphasis on the closure properties of these classes. Further it is shown that any equationally complete inference system for partial conditional specifications may be extended to an inference system for partial higher-order conditional specifications, which is equationally complete w.r.t. the class of all extensional models. Then, applying some previous results, a deduction system is proposed, equationally complete for the class of extensional models of a partial conditional specification. Finally, turning the attention to the special important case of termextensional models, it is first shown a sound and equationally complete inference system and then necessary and sufficient conditions are given for the existence of free models, which are also free in the class of term-generated extensional models.

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 note on two-weight inequalities for multiple Hardy-type operators

2005 ◽  
Vol 3 (3) ◽  
pp. 223-237 ◽  
Author(s):  
Alexander Meskhi
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Hardy Type ◽  
Right Hand ◽  
The Right ◽  
Necessary And Sufficient ◽  
Hardy Type Operators

Necessary and sufficient conditions on a pair of weights guaranteeing two-weight estimates for the multiple Riemann-Liouville transforms are established provided that the weight on the right-hand side satisfies some additional conditions.

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