On the one self-improving property of exponent
Journal Title: Дослідження в математиці і механіці - Year 2015, Vol 20, Issue 2
Abstract
In this note an alternative proof of the well-known property, associated with the so-called “self-improving property”, is presented. This property states that if for some 𝐵 > 1, 1 < 𝑞 < ∞, nonnegative on (0,+∞) function 𝑤 satisfies ∞_ 𝑟 𝑤(𝑥) 𝑥𝑞 𝑑𝑥 6 𝐵 𝑟𝑞 _𝑟 0 𝑤(𝑥) 𝑑𝑥, 𝑟 > 0, then the same inequality remains valid if to reduce the “little” the 𝑞 and to increase the 𝐵. This proof contains the best possible parameters for such a possible “self-improving”. On the use of this property, for example, the proof of the Hardy transformation tightness in the space with the weight, is based. Perhaps the knowledge of the exact values of the parameters of this “self-improving” will be useful for the Hardy operator norm finding in the weighted space, or for the improving t of the norm’s well-known estimates.
Authors and Affiliations
A. Korenovskyi, K. Vrublevska
Keywords
Related Articles
- EP ID EP417172
- DOI -
- Views 88
- Downloads 0
ASYMPTOTICS OF SLOWLY VARYING SOLUTIONS OF SECOND-ORDER DIFFERENTIAL EQUATIONS WITH REGULARLY AND RAPIDLY VARYING NONLINEARITIES.
In this paper for the second-order differential equation which has a right-hand side containing the sum of the terms with regularly and rapidly varying nonlinearities the necessary and sufficient conditions of the existe...
ON THE EXISTENCE OF AN INTEGRAL MANIFOLD OF A SPECIAL TYPE OF A SOME NONLINEAR DIFFERENTIAL SYSTEM
A nonlinear system of the differential equations is considered. This system is a linear extension of the nonlinear differential equation on a circle. The coefficients of the system belong to the class F, which are the fu...
Some remarks on the absolute continuity of set-valued mappings
Recently, many authors have considered the existence, uniqueness and properties of solutions of set-valued differential and integral-differential equations, higher-order equations and investigated impulsive and control s...
Hyperbolic equation with singular coefficient
Dedicated to research existence conditions of quasi-linear differential equations with measurable coefficients, ie the study limitations imposed by the non linearity in which the system will have to research a solution a...
Noninvertible analogue for the congugate mapping for the pair of tent-like mappings
The functional equation which appear in the definition of the topological conjugacy of the tent-like piecewise linear mappings of an interval into itself. All the solutions of this functional equations are obtained and t...