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
Keywords
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- EP ID EP234561
- DOI 10.15393/j3.art.2013.2382
- Views 114
- Downloads 0
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