ANALOG OF AN INEQUALITY OF BOHR FOR INTEGRALS OF FUNCTIONS FROM L p (Rn)
Let p ∈ (2, +∞], n ≥ 1 and ∆ = (∆1, . . . , ∆n), ∆k > 0, 1 ≤ k ≤ n. It is proved that for functions γ(t) ∈ L p (R n ) spectrum of which is separated from each of n the coordinate hyperplanes on the distance not less than ∆k, 1 ≤ k ≤ n respectively, the inequality is valid: ∫ Et γ(τ ) dτ L∞(Rn) ≤...
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Format: | Article |
Language: | English |
Published: |
Petrozavodsk State University
2014-11-01
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Series: | Проблемы анализа |
Subjects: | |
Online Access: | http://issuesofanalysis.petrsu.ru/article/genpdf.php?id=2569&lang=en |
Summary: | Let p ∈ (2, +∞], n ≥ 1 and ∆ = (∆1, . . . , ∆n), ∆k > 0, 1 ≤ k ≤ n. It is proved that for functions γ(t) ∈ L p (R n ) spectrum of which is separated from each of n the coordinate hyperplanes on the distance not less than ∆k, 1 ≤ k ≤ n respectively, the inequality is valid: ∫ Et γ(τ ) dτ L∞(Rn) ≤ C n (q) [∏n k=1 1 ∆ 1/q k ] ∥γ(τ )∥Lp(Rn) , where t = (t1, . . . , tn) ∈ R n , Et = {τ | τ = (τ1, . . . , τn) ∈ R n , τj ∈ [0, tj ], if tj ≥ 0, and τj ∈ [tj , 0], if tj < 0, 1 ≤ j ≤ n}, and the constant C(q) > 0, 1 p + 1 q = 1 does not depend on γ(τ ) and vector ∆. |
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ISSN: | 2306-3424 2306-3432 |