On a generalization of an inequality of Bohr
Journal Title: Проблемы анализа-Issues of Analysis - Year 2013, Vol 2, Issue 2
Abstract
Пусть pϵ(1,2], n≥1, S⊆R n и Г(S,p)- множество всех тех функций, γ(t)ϵL p(R n), носитель преобразования Фурье которых лежит в S. В работе получены условия выполнения неравенства ||∫ E t γ(τ)dτ|| L ∞(R n)≤C||γ(τ)|| L p(R n), где t=(t 1, t 2,..., t n)ϵR n, E t = {τ|τ=(τ 1,τ 2,...,τ n)ϵR n, τ jϵ(t j≤0, 1≤j≤n}, γ(τ)ϵГ(S,p) и константа C не зависит от γ(τ). Также рассмотрены некоторые условия выполнения неравенства на нетривиальных подмножествах Г(S,p) в случаях, когда оно не выполняется на всем Г(S,p). Let pϵ(1; 2]; n≥1; S⊆R n; and Г(S; p)- the set of all functions, γ(t)ϵL p(R n); the support of the Fourier transform of which lies in S. We obtain the inequality conditions ||∫ E t γ(τ)dτ|| L ∞(R n)≤C||γ(τ)|| L p(R n), where t = (t 1; t 2;... ; t n)ϵR n; E t = {τ|τ=(τ 1,τ 2,...,τ n)ϵR n; τ jϵ[0; t j] if t j≥0; and τ jϵ(t j,0]; if t j≥0; 1≤j≤n}, γ(t)ϵГ(S; p) and constant C does not depend on γ(t). Also were considered some validity conditions on the inequality on non-trivial subsets Г(S; p) in cases, where they were not satisfied on the whole Г(S; p).
Authors and Affiliations
B. F. Ivanov
MONOTONICITY AND CONVEXITY PROPERTIES OF THE NIELSEN’S β-FUNCTION
The Nielsen’s β-function provides a powerful tool for evaluating and estimating certain integrals, series and mathematical constants. It is related to other special functions such as the digamma function, the Euler’s bet...
ОЦЕНКИ РАЗНОСТЕЙ РЕШЕНИЙ НЕЛИНЕЙНЫХ ИНТЕГРАЛЬНЫХ УРАВНЕНИЙ В БАНАХОВОМ ПРОСТРАНСТВЕ
In this paper we consider estimations of difference of nonlinear integral equations in Banach space.
О ВЕСЕ НИГДЕ НЕ ПЛОТНЫХ ПОДМНОЖЕСТВ
In this paper we study the nd-weight of topological spaces. The relations between nd-weight, weight, density and cellularity are studied. The estimates for the nd-weight of the sum and the product of topological spaces a...
WILKER AND HUYGENS-TYPE INEQUALITIES INVOLVING GUDERMANNIAN AND THE INVERSE GUDERMANNIAN FUNCTIONS
Five Wilker Huygens-type inequalities involving Gudermannian and the inverse Gudermannian functions are obtained. The Schwab-Borchardt mean plays a crucial role in the proofs. Also, an analytical inequality for the sums...
ON METRIC SPACE VALUED FUNCTIONS OF BOUNDED ESSENTIAL VARIATION
Let ∅≠T ⊂ R and let X be a metric space. For an ideal J ⊂ P(T) and a function f:T-> X, we define the essential variation V^J ess(f, T) as the in mum of all variations V (g; T) where g:T-> X, g = f on T\E, and E in J. We...