scholarly journals Some new multidimensional hardy-type inequalities with kernels via convexity

2013 ◽  
Vol 93 (107) ◽  
pp. 153-164
Author(s):  
James Oguntuase ◽  
Philip Durojaye
Keyword(s):  
Dimensional Case ◽  
One Dimensional ◽  
Hardy Type Inequalities ◽  
Hardy Type ◽  
The One ◽  
Positive Kernels ◽  
Hardy Type Operators

We prove some new multidimensional Hardy-type inequalities involving general Hardy type operators with positive kernels for functions ? which may not necessarily be convex but satisfy the condition A?(x) ? ?(x) ? B? (x), where ? is convex. Our approach is mainly the use of convexity argument and the results obtained are new even for the one-dimensional case and also unify and extend several inequalities of Hardy type known in the literature.

scholarly journals MULTIDIMENSIONAL HARDY-TYPE INEQUALITIES VIA CONVEXITY

2008 ◽  
Vol 77 (2) ◽  
pp. 245-260 ◽  
Author(s):  
JAMES A. OGUNTUASE ◽  
LARS-ERIK PERSSON ◽  
ALEKSANDRA ČIŽMEŠIJA
Keyword(s):  
Positive Function ◽  
Dimensional Case ◽  
One Dimensional ◽  
Hardy Type Inequalities ◽  
Left Hand ◽  
General Integral ◽  
Hardy Type ◽  
Almost Everywhere ◽  
Power Weights ◽  
The One

AbstractLet an almost everywhere positive function Φ be convex forp>1 andp<0, concave forp∈(0,1), and such thatAxp≤Φ(x)≤Bxpholds on$\mathbb {R}_{+}$for some positive constantsA≤B. In this paper we derive a class of general integral multidimensional Hardy-type inequalities with power weights, whose left-hand sides involve$\Phi ( \int _0^{x_1} \cdots \int _0^{x_n} f(\t )\,d\t )$instead of$[ \int _0^{x_1} \cdots \int _0^{x_n} f(\t ) \, d\t ]^p$, while the corresponding right-hand sides remain as in the classical Hardy’s inequality and have explicit constants in front of integrals. We also prove the related dual inequalities. The relations obtained are new even for the one-dimensional case and they unify and extend several inequalities of Hardy type known in the literature.

One-dimensional Hardy-type inequalities in many dimensions

1998 ◽  
Vol 128 (4) ◽  
pp. 833-848 ◽  
Author(s):  
Gord Sinnamon
Keyword(s):  
Integral Operators ◽  
Weighted Norm ◽  
Weighted Inequalities ◽  
Weighted Norm Inequalities ◽  
Averaging Operators ◽  
One Dimensional ◽  
Hardy Type Inequalities ◽  
Hardy Type ◽  
Higher Dimensional ◽  
The One

Weighted inequalities for certain Hardy-type averaging operators in are shown to be equivalent to weighted inequalities for one-dimensional operators. Known results for the one-dimensional operators are applied to give weight characterisations, with best constants in some cases, in the higher-dimensional setting. Operators considered include averages over all dilations of very general starshaped regions as well as averages over all balls touching the origin. As a consequence, simple weight conditions are given which imply weighted norm inequalities for a class of integral operators with monotone kernels.

Regions-based Two-dimensional Continua: The Euclidean Case

2018 ◽  
Author(s):  
Geoffrey Hellman ◽  
Stewart Shapiro
Keyword(s):  
Mathematical Induction ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Entire Space ◽  
Euclidean Case ◽  
Free Basis ◽  
One Dimensional ◽  
Archimedean Property ◽  
The One

This chapter develops a Euclidean, two-dimensional, regions-based theory. As with the semi-Aristotelian account in Chapter 2, the goal here is to recover the now orthodox Dedekind–Cantor continuum on a point-free basis. The chapter derives the Archimedean property for a class of readily postulated orientations of certain special regions, what are called “generalized quadrilaterals” (intended as parallelograms), by which the entire space is covered. Then the chapter generalizes this to arbitrary orientations, and then establishes an isomorphism between the space and the usual point-based one. As in the one-dimensional case, this is done on the basis of axioms which contain no explicit “extremal clause”, and we have no axiom of induction other than ordinary numerical (mathematical) induction.

scholarly journals Preserving the Shape of Functions by Applying Multidimensional Schoenberg-Type Operators

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1016
Author(s):  
Camelia Liliana Moldovan ◽  
Radu Păltănea
Keyword(s):  
Applied Sciences ◽  
Degree Of Approximation ◽  
Higher Order ◽  
Second Order ◽  
Dimensional Case ◽  
Moduli Of Continuity ◽  
One Dimensional ◽  
Multidimensional Generalization ◽  
The One

The paper presents a multidimensional generalization of the Schoenberg operators of higher order. The new operators are powerful tools that can be used for approximation processes in many fields of applied sciences. The construction of these operators uses a symmetry regarding the domain of definition. The degree of approximation by sequences of such operators is given in terms of the first and the second order moduli of continuity. Extending certain results obtained by Marsden in the one-dimensional case, the property of preservation of monotonicity and convexity is proved.

scholarly journals Limit of 𝑝-Laplacian obstacle problems

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Raffaela Capitanelli ◽  
Maria Agostina Vivaldi
Keyword(s):  
Asymptotic Behavior ◽  
Uniform Convergence ◽  
Explicit Expression ◽  
Positive Measure ◽  
Sufficient Conditions ◽  
Obstacle Problems ◽  
Dimensional Case ◽  
Asymptotic Behavior Of Solutions ◽  
One Dimensional ◽  
The One

AbstractIn this paper, we study asymptotic behavior of solutions to obstacle problems for p-Laplacians as {p\to\infty}. For the one-dimensional case and for the radial case, we give an explicit expression of the limit. In the n-dimensional case, we provide sufficient conditions to assure the uniform convergence of the whole family of the solutions of obstacle problems either for data f that change sign in Ω or for data f (that do not change sign in Ω) possibly vanishing in a set of positive measure.

Total progeny in a multitype critical age dependent branching process with immigration

1974 ◽  
Vol 11 (3) ◽  
pp. 458-470 ◽  
Author(s):  
Howard J. Weiner
Keyword(s):  
Limit Theorem ◽  
Branching Process ◽  
Cell Types ◽  
Dimensional Case ◽  
One Dimensional ◽  
Age Dependent ◽  
Total Progeny ◽  
Critical Age ◽  
The One ◽  
Branching Process With Immigration

In a multitype critical age dependent branching process with immigration, the numbers of cell types born by t, divided by t2, tends in law to a one-dimensional (degenerate) law whose Laplace transform is explicitily given. The method of proof makes a correspondence between the moments in the m-dimensional case and the one-dimensional case, for which the corresponding limit theorem is known. Other applications are given, a possible relaxation of moment assumptions, and extensions are indicated.

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