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

## 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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## References

- 1.N. Bourbaki,
*Integration. Measures, Integration of Measures*(Nauka, Moscow, 1967) [in Russian].zbMATHGoogle Scholar - 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.I. M. Gel’fand and G. E. Shilov,
*Generalized Functions*, Vol. 1:*Properties and Operations*(Fizmatlit, Moscow, 1959; Academic, New York, 1964).zbMATHGoogle Scholar - 4.
*Functional Analysis*, Ed. by S. G. Krein (Nauka, Moscow, 1972; Wolters-Noordhoff, Groningen, 1972).Google Scholar - 5.E. M. Stein and G. Weiss,
*Introduction to Fourier Analysis on Euclidean Spaces*(Princeton Univ. Press, Princeton, NJ, 1971; Mir, Moscow, 1974).zbMATHGoogle Scholar