Abstract :Let p ∈ (2; +∞], n ≥ 1 and ∆ = (∆1, ... , ∆n), ∆K>0,1≤k≤n. It is proved that for functions Ɣ (t)∈Lp(Rn) 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∫Ɣ(Ƭ)d\Ƭ||L∞(Rn) Cn(q)[∏nk=1 1/∆1/qk]∥Ɣ(Ƭ)∥ Lp(Rn) , where t = (t 1 , ... ,t n )∈R n , E t ={Ƭ|Ƭ=(Ƭ 1 , ... ,Ƭ n )∈R n , Ƭ j ∈[0,t j ], if t j ≥0, and Ƭ j ∈[t j ,0], if t j<0,1≤k≤n}, and the constant C(q)>0, 1/p+1/q= 1 does not depend on Ɣ(Ƭ) and vector ∆.