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
ON SOME HARMONIC FUNCTIONS RELATED TO HOLOMORPHIC FUNCTIONS WITH A POSITIVE REAL PART
In the paper we examine some holomorphic functions and complex harmonic functions, which satisfy certain conditions of a Mocanu kind. We also consider their relations with appropriate coefficient conditions. The paper is...
THE SCHWARZIAN DERIVATIVES OF HARMONIC FUNCTIONS AND UNIVALENCE CONDITIONS
In the paper we obtain some analogues of Nehari’s univalence conditions for sense-preserving functions that are harmonic in the unit disc D = {z ∈ C : |z| < 1}.
ON THE INEQUALITIES FOR THE VOLUME OF THE UNIT BALL Ω_N IN R^N
The inequalities about the volume of the unit ball Ω_n in R^n were studies by several authors, especially Horst Alzer has a great contribution to this topic. Thereafter many authors produced numerous papers on this topic...
BOUNDARY VALUE PROBLEMS FOR INTEGRAL EQUATIONS WITH OPERATOR MEASURES
We consider integral equations with operator measures on a segment in the infinite-dimensional case. These measures are defined on Borel sets of the segment and take values in the set of linear bounded operators acting i...
REDUCED p-MODULUS, p-HARMONIC RADIUS AND p-HARMONIC GREEN’S MAPPINGS
We consider the definitions and properties of the metric characteristics of the spatial domains previously introduced by the author, and their connection with the class of mappings, the particular case of which are the h...