Supplement to Hölder’s inequality. II

Abstract

Suppose that m ≥ 2, numbers p₁, …, p_m ∈ (1, +∞] satisfy the inequality \(\frac{1}{{{p_1}}} + \cdots + \frac{1}{{{p_m}}} < 1\), and functions \({\gamma _1} \in {L^{{p_1}}}\left( {{ℝ^1}} \right), \cdots ,{\gamma _m} \in {L^{{p_m}}}\left( {{ℝ^1}} \right)\) are given. It is proved that if the set of “resonance” points of each of these functions is nonempty and the “nonresonance” condition holds (both notions were defined by the author for functions in L^p(ℝ¹), p ∈ (1, +∞]), then \(\mathop {\sup }\limits_{a,b \in {R^1}} \left| {\mathop \smallint \limits_a^b \prod\limits_{k = 1}^m {[{\gamma _k}\left( \tau \right) + \Delta {\gamma _k}\left( \tau \right)]} d\tau } \right| \leqslant C\prod\limits_{k = 1}^m {{{\left\| {{\gamma _k} + \Delta {\gamma _k}} \right\|}_{L_{ak}^{pk}\left( {{R^1}} \right)}}} \) where the constant C > 0 is independent of the functions \(\Delta {\gamma _k} \in L_{ak}^{pk}\left( {{ℝ^1}} \right)\)
and \(L_{ak}^{pk}\left( {{ℝ^1}} \right) \subset {L^{pk}}\left( {{ℝ^1}} \right)\), 1 ≤ k ≤ m, are special normed spaces. A condition for the integral over ℝ¹ of a product of functions to be bounded is also given.