Inequalities with Sharp Constants for Operators Satisfying the Conditions of p-Convexity or p-Concavity Type

2000 ◽  
pp. 1179-1186
Author(s):  
T. V. Tararykova
Keyword(s):  
Sharp Constants

General sharp weighted Caffarelli–Kohn–Nirenberg inequalities

2018 ◽  
Vol 149 (03) ◽  
pp. 691-718 ◽  
Author(s):  
Nguyen Lam
Keyword(s):  
Mass Transport ◽  
Sharp Constants ◽  
Optimal Mass Transport

AbstractIn this paper, we will use optimal mass transport combining with suitable transforms to study the sharp constants and optimizers for a class of the Gagliardo–Nirenberg and Caffarelli–Kohn–Nirenberg inequalities. Moreover, we will investigate these inequalities with and without the monomial weights $x_{1}^{A_{1}} \cdots x_{N}^{A_{N}}$ on ℝN.

SHARP POINCARÉ–SOBOLEV INEQUALITIES AND THE SHORTEST LENGTH OF SIMPLE CLOSED GEODESICS ON A TOPOLOGICAL TWO SPHERE

2004 ◽  
Vol 06 (05) ◽  
pp. 781-792 ◽  
Author(s):  
MEIJUN ZHU
Keyword(s):  
Riemannian Manifold ◽  
Sobolev Inequalities ◽  
Square Root ◽  
Two Dimensional ◽  
Closed Geodesics ◽  
Sharp Constants ◽  
Simple Closed Geodesics

We show that the sharp constants of Poincaré–Sobolev inequalities for any smooth two dimensional Riemannian manifold are less than or equal to [Formula: see text]. For a smooth topological two sphere M2, the sharp constants are [Formula: see text] if and only if M2 is isometric to two sphere S2 with the standard metric. In the same spirit, we show that for certain special smooth topological sphere the ratio between the shortest length of simple closed geodesics and the square root of its area is less than or equals to [Formula: see text].

scholarly journals Bellman function approach to the sharp constants in uniform convexity

2018 ◽  
Vol 11 (1) ◽  
pp. 89-93
Author(s):  
Paata Ivanisvili
Keyword(s):  
Uniform Convexity ◽  
Function Approach ◽  
Function Technique ◽  
Bellman Function ◽  
Sharp Constants

AbstractWe illustrate a Bellman function technique in finding the modulus of uniform convexity of {L^{p}} spaces.

On Asymptotics of the Sharp Constants of the Markov–Bernshtein Inequalities for the Sobolev Spaces

2018 ◽  
Vol 39 (5) ◽  
pp. 609-622 ◽  
Author(s):  
A. I. Aptekarev ◽  
A. Draux ◽  
D. N. Tulyakov
Keyword(s):  
Sobolev Spaces ◽  
Sharp Constants

scholarly journals Towards sharp Bohnenblust–Hille constants

2018 ◽  
Vol 20 (03) ◽  
pp. 1750029 ◽  
Author(s):  
Daniel Pellegrino ◽  
Eduardo V. Teixeira
Keyword(s):  
Exponential Growth ◽  
Optimization Problems ◽  
Best Constants ◽  
Sharp Constants ◽  
Linear Forms ◽  
Entropy Formula ◽  
Optimal Constants

We investigate the optimality problem associated with the best constants in a class of Bohnenblust–Hille-type inequalities for [Formula: see text]-linear forms. While germinal estimates indicated an exponential growth, in this work we provide strong evidences to the conjecture that the sharp constants in the classical Bohnenblust–Hille inequality are universally bounded, irrespectively of the value of [Formula: see text]; hereafter referred as the Universality Conjecture. In our approach, we introduce the notions of entropy and complexity, designed to measure, to some extent, the complexity of such optimization problems. We show that the notion of entropy is critically connected to the Universality Conjecture; for instance, that if the entropy grows at most exponentially with respect to [Formula: see text], then the optimal constants of the [Formula: see text]-linear Bohnenblust–Hille inequality for real scalars are indeed bounded universally with respect to [Formula: see text]. It is likely that indeed the entropy grows as [Formula: see text], and in this scenario, we show that the optimal constants are precisely [Formula: see text]. In the bilinear case, [Formula: see text], we show that any extremum of the Littlewood’s [Formula: see text] inequality has entropy [Formula: see text] and complexity [Formula: see text], and thus we are able to classify all extrema of the problem. We also prove that, for any mixed [Formula: see text]-Littlewood inequality, the entropy do grow exponentially and the sharp constants for such a class of inequalities are precisely [Formula: see text]. In addition to the notions of entropy and complexity, the approach we develop in this work makes decisive use of a family of strongly non-symmetric [Formula: see text]-linear forms, which has further consequences to the theory, as we explain herein.

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