A Supplement to Hölder’s Inequality. The Resonance Case. II
Boris F. Ivanov · Vestnik St Petersburg University Mathematics · 2018
Suppose that m ≥ 2, and numbers p1, …, p m ∈ (1, +∞] satisfy the inequality 1/p1+…+1/p m < 1, and functions $${\gamma _1} \in {L^{{p_1}}}\left( {{\mathbb{R}^1}} \right),...,{\gamma _m} \in {L^{{p_m}}}\left( {{\mathbb{R}^1}} \right)$$ are given. It is proven that, if the set of resonance points of each of these functions is nonempty and the so-called resonance condition holds, there will always exist arbitrarily small (in norm) perturbations $$\Delta {\gamma _k} \in {L^{{p_k}}}\left( {{\mathbb{R}^1}} \right)$$ under which the set of resonance points of the function γk + Δγk coincides with that of the function γ k for 1 ≤ k ≤ m, but in this case, $${\left\| {\int\limits_0^t {\prod\limits_{k = 1}^m {[{\gamma _k}\left( \tau \right) + \Delta {\gamma _k}\left( \tau \right)]d\tau } } } \right\|_{{L^\infty }\left( {{\mathbb{R}^1}} \right)}} = \infty$$ The notion of a resonance point and the resonance condition for the functions of the spaces L p (R1), p ∈ (1, +∞], were introduced by the author in his previous papers.