# A Supplement to Hölder’s Inequality. The Resonance Case. I

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## Abstract

Suppose that *m* ≥ 2, numbers *p*_{1}, …, *p*_{m} ∈ (1, +∞] satisfy the inequality \(\frac{1}{{{p_1}}} + ... + \frac{1}{{{p_m}}} < 1\), and functions γ_{1} ∈ \({L^{{p_1}}}\)(ℝ^{1}), …, γ_{m} ∈ \({L^{{p_m}}}\)(ℝ^{1}) are given. It is proved that if the set of “resonance points” of each of these functions is nonempty and the so-called “resonance condition” holds, then there are arbitrarily small (in norm) perturbations Δγ_{k} ∈ \({L^{{p_k}}}\)(ℝ^{1}) under which the resonance set of each function γ_{k} + Δγ_{k} coincides with that of γ_{k} for 1 ≤ *k* ≤ *m*, but \({\left\| {\int\limits_0^t {\prod\limits_{k = 0}^m {\left[ {{\gamma _k}\left( \tau \right) + \Delta {\gamma _k}\left( \tau \right)} \right]d\tau } } } \right\|_{{L^\infty }\left( {{\mathbb{R}^1}} \right)}} = \infty \). The notion of a resonance point and the resonance condition for functions in the spaces *L*^{p}(ℝ^{1}), *p* ∈ (1, +∞], were introduced by the author in his previous papers.

## Keywords

Hölder's inequality## Preview

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