scholarly journals Sobolev mappings and moduli inequalities on Carnot groups

2020 ◽  
Vol 17 (2) ◽  
pp. 215-233
Author(s):  
Evgenii Sevost'yanov ◽  
Alexander Ukhlov
Keyword(s):  
Integrable Function ◽  
Sobolev Class ◽  
Carnot Groups ◽  
Inverse Mapping ◽  
Finite Distortion ◽  
Inverse Mapping Theorem ◽  
Sobolev Mappings ◽  
Locally Integrable Function ◽  
Moduli Inequalities

We study the mappings that satisfy moduli inequalities on Carnot groups. We prove that the homeomorphisms satisfying the moduli inequalities ($Q$-homeomor\-phisms) with a locally integrable function $Q$ are Sobolev mappings. On this base in the frameworks of the weak inverse mapping theorem, we prove that, on the Carnot groups $\mathbb G,$ the mappings inverse to Sobolev homeomorphisms of finite distortion of the class $W^1_{\nu,\loc}(\Omega;\Omega')$ belong to the Sobolev class $W^1_{1,\loc}(\Omega';\Omega)$.

scholarly journals Sobolev Mappings and Moduli Inequalities on Carnot Groups

2020 ◽  
Vol 249 (5) ◽  
pp. 754-768
Author(s):  
Evgenii A. Sevost’yanov ◽  
Alexander Ukhlov
Keyword(s):  
Carnot Groups ◽  
Sobolev Mappings ◽  
Moduli Inequalities

A concept of inner prederivative for set-valued mappings and its applications

2018 ◽  
Vol 24 (3) ◽  
pp. 1059-1074
Author(s):  
Michel H. Geoffroy ◽  
Yvesner Marcelin
Keyword(s):  
Optimality Conditions ◽  
Necessary Optimality Conditions ◽  
Welfare Economics ◽  
Inverse Mapping ◽  
First Order ◽  
Inverse Mapping Theorem ◽  
Homogeneous Set ◽  
Optimal Allocations ◽  
Variational Perturbations ◽  
Set Valued Mappings

We introduce a class of positively homogeneous set-valued mappings, called inner prederivatives, serving as first order approximants to set-valued mappings. We prove an inverse mapping theorem involving such prederivatives and study their stability with respect to variational perturbations. Then, taking advantage of their properties we establish necessary optimality conditions for the existence of several kind of minimizers in set-valued optimization. As an application of these last results, we consider the problem of finding optimal allocations in welfare economics. Finally, to emphasize the interest of our approach, we compare the notion of inner prederivative to the related concepts of set-valued differentiation commonly used in the literature.

scholarly journals TOWARDS DIFFERENTIAL CALCULUS IN STRATIFIED GROUPS

2013 ◽  
Vol 95 (1) ◽  
pp. 76-128 ◽  
Author(s):  
VALENTINO MAGNANI
Keyword(s):  
Implicit Function Theorem ◽  
Group Structure ◽  
Mean Value ◽  
Implicit Function ◽  
Inverse Mapping ◽  
Natural Group ◽  
Stratified Groups ◽  
First Order ◽  
Inverse Mapping Theorem ◽  
Rank Theorem

AbstractWe study graded group-valued continuously differentiable mappings defined on stratified groups, where differentiability is understood with respect to the group structure. We characterize these mappings by a system of nonlinear first-order PDEs, establishing a quantitative estimate for their difference quotient. This provides us with a mean value estimate that allows us to prove both the inverse mapping theorem and the implicit function theorem. The latter theorem also relies on the fact that the differential admits a proper factorization of the domain into a suitable inner semidirect product. When this splitting property of the differential holds in the target group, then the inverse mapping theorem leads us to the rank theorem. Both implicit function theorem and rank theorem naturally introduce the classes of image sets and level sets. For commutative groups, these two classes of sets coincide and correspond to the usual submanifolds. In noncommutative groups, we have two distinct classes of intrinsic submanifolds. They constitute the so-called intrinsic graphs, that are defined with respect to the algebraic splitting and everywhere possess a unique metric tangent cone equipped with a natural group structure.

The Weak Inverse Mapping Theorem

2015 ◽  
Vol 34 (3) ◽  
pp. 321-342 ◽  
Author(s):  
Daniel Campbell ◽  
Stanislav Hencl ◽  
František Konopecký
Keyword(s):  
Inverse Mapping ◽  
Inverse Mapping Theorem

Oscillation and Nonoscillation Criteria for a Second Order Linear Equation

1999 ◽  
Vol 6 (5) ◽  
pp. 401-414
Author(s):  
T. Chantladze ◽  
N. Kandelaki ◽  
A. Lomtatidze
Keyword(s):  
Linear Equation ◽  
Integrable Function ◽  
Second Order ◽  
Order Linear ◽  
Locally Integrable Function ◽  
Oscillation And Nonoscillation Criteria ◽  
Oscillation And Nonoscillation

Abstract New oscillation and nonoscillation criteria are established for the equation 𝑢″ + 𝑝(𝑡)𝑢 = 0, where 𝑝 : ]1, + ∞[ → 𝑅 is the locally integrable function. These criteria generalize and complement the well known criteria of E. Hille, Z. Nehari, A. Wintner, and P. Hartman.

Scattering Length and the Spectrum of –Δ + V

2006 ◽  
Vol 49 (1) ◽  
pp. 144-151 ◽  
Author(s):  
Michael Taylor
Keyword(s):  
Discrete Spectrum ◽  
Integrable Function ◽  
Scattering Length ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Locally Integrable Function ◽  
Necessary And Sufficient

AbstractGiven a non-negative, locally integrable functionV on ℝn, we give a necessary and sufficient condition that –Δ + V have purely discrete spectrum, in terms of the scattering length ofV restricted to boxes.

scholarly journals BEST CONSTANTS IN THE WEAK-TYPE ESTIMATES FOR UNCENTERED MAXIMAL OPERATORS

2012 ◽  
Vol 54 (3) ◽  
pp. 655-663
Author(s):  
ADAM OSȨKOWSKI
Keyword(s):  
Maximal Operator ◽  
Borel Measure ◽  
Weak Type ◽  
Integrable Function ◽  
Maximal Operators ◽  
Weak Type Estimates ◽  
Locally Integrable Function ◽  
Image Position ◽  
Strong Maximal Operator ◽  
General Product

AbstractLet μ be a Borel measure on ℝ. The paper contains the proofs of the estimates and Here A is a subset of ℝ, f is a μ-locally integrable function, μ is the uncentred maximal operator with respect to μ and cp,q, and Cp,q are finite constants depending only on the parameters indicated. In the case when μ is the Lebesgue measure, the optimal choices for cp,q and Cp,q are determined. As an application, we present some related tight bounds for the strong maximal operator on ℝn with respect to a general product measure.

scholarly journals Differential calculus in Fréchet spaces

1981 ◽  
Vol 24 (1) ◽  
pp. 93-122 ◽  
Author(s):  
Duong Minh Duc
Keyword(s):  
Differential Calculus ◽  
Fréchet Spaces ◽  
Inverse Mapping ◽  
Inverse Mapping Theorem ◽  
Special Case

We apply Keller's method to the study of differential calculus in Frechet spaces and establish an inverse mapping theorem. A special case of this theorem is similar to a theorem of Yamamuro.

scholarly journals An extension of a Hardy-Littlewood-Pólya inequality

1978 ◽  
Vol 21 (1) ◽  
pp. 11-15 ◽  
Author(s):  
A. Erdélyi
Keyword(s):  
Integrable Function ◽  
Right Hand ◽  
Usual Norm ◽  
Locally Integrable Function ◽  
Image Position ◽  
The Right

The Hardy-Littlewood-Pólya inequality in question can be written in the formHere and throughout, all functions are assumed to be locally integrable on ]0,∞[, 1≤p≤∞,p-1+(p′)-1=1 (with similar conventions for q,r,s), is the usual norm on Lp(0,∞), and if the right hand side is finite, then (1.1) is understood to mean thatdefines a locally integrable function Kf for which (1.1) holds.

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