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

Mathematics
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Abstract

Suppose that m ≥ 2, numbers p1, …, p m ∈ (1, +∞] satisfy the inequality \(\frac{1}{{{p_1}}} + ... + \frac{1}{{{p_m}}} < 1\), and functions γ1\({L^{{p_1}}}\)(ℝ1), …, γ m \({L^{{p_m}}}\)(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 L p (ℝ1), p ∈ (1, +∞], were introduced by the author in his previous papers.

Keywords

Hölder's inequality 

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References

  1. 1.
    N. Bourbaki, Integration. Measures, Integration of Measures (Nauka, Moscow, 1967) [in Russian].MATHGoogle Scholar
  2. 2.
    B. F. Ivanov, “On the Hölder inequality,” in Complex Analysis and Applications, Proc. 8th Int. Conf., Petrozavodsk, June 3–9, 2016 (Petrozavodsk. Gos. Univ., Petrozavodsk, 2016), pp. 31–35.Google Scholar
  3. 3.
    I. M. Gel’fand and G. E. Shilov, Generalized Functions, Vol. 1: Properties and Operations (Fizmatlit, Moscow, 1959; Academic, New York, 1964).MATHGoogle Scholar
  4. 4.
    Functional Analysis, Ed. by S. G. Krein (Nauka, Moscow, 1972; Wolters-Noordhoff, Groningen, 1972).Google Scholar
  5. 5.
    E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces (Princeton Univ. Press, Princeton, NJ, 1971; Mir, Moscow, 1974).MATHGoogle Scholar

Copyright information

© Allerton Press, Inc. 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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