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

  • B. F. IvanovEmail author


Suppose that m ≥ 2, numbers p1, …, pm ∈ (1, +∞] satisfy the inequality \(\frac{1}{{{p_1}}} + ... + \frac{1}{{{p_m}}} < 1\), and functions γ1\({L^{{p_1}}}\)(ℝ1), …, γm\({L^{{p_m}}}\)(ℝ1) 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 Δγk\({L^{{p_k}}}\)(ℝ1) under which the resonance set of each function γk + Δγk coincides with that of γk for 1 ≤ km, 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 Lp(ℝ1), p ∈ (1, +∞], were introduced by the author in his previous papers.


Hölder's inequality 


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© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.St. Petersburg State University of Industrial Technologies and DesignHigher School of Technology and EnergySt. PetersburgRussia

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