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 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 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.

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.

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.

Weighted higher order exponential type inequalities in metric spaces and applications

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Huiju Wang ◽  
Pengcheng Niu
Keyword(s):  
Homogeneous Space ◽  
Lie Groups ◽  
Sobolev Spaces ◽  
Exponential Type ◽  
Metric Spaces ◽  
Type Inequality ◽  
Higher Order ◽  
Geodesic Space ◽  
Stratified Lie Groups ◽  
Weighted Case

AbstractIn this paper, we establish weighted higher order exponential type inequalities in the geodesic space {({X,d,\mu})} by proposing an abstract higher order Poincaré inequality. These are also new in the non-weighted case. As applications, we obtain a weighted Trudinger’s theorem in the geodesic setting and weighted higher order exponential type estimates for functions in Folland–Stein type Sobolev spaces defined on stratified Lie groups. A higher order exponential type inequality in a connected homogeneous space is also given.

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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