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A supplement to Hölder’s inequality. The resonance case. II. (English. Russian original) Zbl 1400.26050

Vestn. St. Petersbg. Univ., Math. 51, No. 2, 124-132 (2018); translation from Vestn. St-Peterbg. Univ., Ser. I, Mat. Mekh. Astron. 5(63), No. 2, 194-204 (2018).
Summary: Suppose that \(m\geq 2\), and numbers \(p_1, \dots, p_m\in(1, +\infty]\) satisfy the inequality \(\frac{1}{p_1}+\cdots+\frac{1}{p_m} < 1\), and functions \(\gamma_1 \in L^{p_1}(\mathbb{R}^1),\dots,\gamma_m \in L^{p_m}(\mathbb{R}^1)\) 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 \(\gamma_k + \Delta\gamma_k\) coincides with that of the function \(\gamma_k\) for \(1\leq k\leq m\), 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 \(L^p(R^1)\), \(p\in(1, +\infty]\), were introduced by the author in his previous papers.
For Part I see [the author, Vestn. St. Petersbg. Univ., Math. 51, No. 1, 49–56 (2018; Zbl 1400.26049); translation from Vestn. St-Peterbg. Univ., Ser. I, Mat. Mekh. Astron. 5(63), No. 1, 65–73 (2018)].

MSC:

26D15 Inequalities for sums, series and integrals

Citations:

Zbl 1400.26049
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References:

[1] Ivanov, B. F., A supplement to Hölder’s inequality. the resonance case. I, Vestn. St. Petersburg Univ.: Math., 51, 49-56, (2018) · Zbl 1400.26049 · doi:10.3103/S1063454118010041
[2] N. Bourbaki, Integration. Measures, Integration of Measures (Nauka, Moscow, 1967) [in Russian]. · Zbl 0156.06001
[3] I. M. Gel’fand and G. E. Shilov, Generalized Functions, Vol. 1: Properties and Operations (Fizmatlit, Moscow, 1959; Academic, New York, 1964). · Zbl 0091.11102
[4] Functional Analysis, Ed. by S. G. Krein (Nauka, Moscow, 1972; Wolters-Noordhoff, Groningen, 1972).
[5] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces (Princeton Univ. Press, Princeton, NJ, 1971; Mir, Moscow, 1974). · Zbl 0232.42007
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