On the Bohr Inequality for the Integrals of Functions from $${{L}^{p}}({{\mathbb{R}}^{n}})$$ with 2 < p < +∞

Let p ∈ (2, +∞), n ≥ 1, S be an open subset of $${{\mathbb{R}}^{n}}$$ , and Γ(S, p) be a set of all the functions γ ∈ Lp( $${{\mathbb{R}}^{n}}$$ ), the spectrum of which belongs to S. It is assumed that for n = 1 the set $$\bar {S}$$ may contain zero, while for n > 1 it may intersect the coordinate hyperplanes. We obtain the sufficient condition validity of the inequality $${{\left\| {\int\limits_{{{E}_{t}}} {\gamma (\tau )d\tau } } \right\|}_{{{{L}^{\infty }}({{\mathbb{R}}^{n}})}}} \leqslant C(n,p,S){{\left\| {\gamma (\tau )} \right\|}_{{{{L}^{p}}({{\mathbb{R}}^{n}})}}},$$ where t = (t1, …, tn) ∈ $${{\mathbb{R}}^{n}}$$ , Et = {τ|τ = (τ1, …, τn) ∈ $${{\mathbb{R}}^{n}}$$ , 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).