On the inequality of Bohr for integrals of functions from Lp(Rn), 2 < p < +∞

Abstract

Let p ∈ (2,+∞), n ≥ 1, S be an open subset of Rn, and Γ(S, p) be a set of all the functions γ ∈ Lp(Rn) spectrum of which belongs to S. If n = 1 then S can contain zero and if n > 1 S can intersect coordinate hyperplanes. It is obtained sufficient condition validity of the inequality

Et γ(τ) dτ

L∞(Rn) ≤ C(n, p, S) γ(τ) Lp(Rn), where t = (t1, . . . , tn) ∈ Rn, Et = {τ|τ = (τ1, . . . , τn) ∈ Rn, τj ∈ [0, tj ], if tj ≥ 0, and τj ∈ [tj , 0], if tj < 0, 1 ≤ j ≤ n}, and the constant C(n, p, S) > 0 does not depend on γ ∈ Γ(S, p). Refs 14.