On one complement to the Hölder inequality: I

Let m ≥ 2, the numbers p1,…, pm ∈ (1, +∞] satisfy the inequality 1p1+..1pm<1, and γ1 ∈ Lp1(ℝ1), …, γm ∈ Lpm(ℝ1). We prove that, if the set of “resonance” points of each of these functions is nonempty and the “nonresonance” condition holds (both concepts have been introduced by the author for functions of spaces Lp(ℝ1), p ∈ (1, +∞]), we have the inequality supa,b∈R1|∫ab∏k=1m[γk(τ)+Δγk(τ)]dτ|≤C∏k=1m‖γk+Δγk‖Lakpk(ℝ1), where the constant C > 0 is independent of functions Δγk∈Lakpk(ℝ1) and Lakpk(ℝ1)⊂Lpk(ℝ1), 1 ≤ k ≤ m are some specially constructed normed spaces. In addition, we give a boundedness condition for the integral of the product of functions over a subset of ℝ1. © 2017, Allerton Press, Inc.