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 Izv. RAN. Ser. Mat., 2006, Volume 70, Issue 3, Pages 129–166 (Mi izv704)

$L^p$-Fourier multipliers with bounded powers

V. V. Lebedeva, A. M. Olevskiib

a Moscow State Institute of Electronics and Mathematics (Technical University)
b Tel Aviv University, School of Mathematical Sciences

Abstract: We consider the space $M_p(\mathbb R^d)$ of $L^p$-Fourier multipliers and give a detailed proof of the following result announced by the authors in $\lbrack10\rbrack$: if $\varphi\colon\mathbb R^d\to \lbrack0, 2\pi\lbrack$ is a measurable function and $\|e^{in\varphi}\|_{M_p}=O(1)$, $n\in\mathbb Z$, for some $p\ne 2$, then the function $\varphi$ is linear in domains complementary to some closed set $E(\varphi)$ of Lebesgue measure zero, and the set of values of the gradient of $\varphi$ is finite. We also consider the question of which sets can appear as $E(\varphi)$. We study the behaviour of the norms of the exponential functions $e^{i\lambda\varphi}$ in the case when the frequency $\lambda$ tends to infinity along a sequence of real numbers. In particular, we construct a homeomorphism $\varphi$ of the line $\mathbb R$ which is non-linear on every interval and satisfies $\|e^{i2^n\varphi}\|_{M_p(\mathbb R)}=O(1)$, $n=0, 1, 2,…$, for all $p$, $1<p<\infty$.

DOI: https://doi.org/10.4213/im704

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English version:
Izvestiya: Mathematics, 2006, 70:3, 549–585

Bibliographic databases:

UDC: 517.51+513.88
MSC: 42A45

Citation: V. V. Lebedev, A. M. Olevskii, “$L^p$-Fourier multipliers with bounded powers”, Izv. RAN. Ser. Mat., 70:3 (2006), 129–166; Izv. Math., 70:3 (2006), 549–585

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/izv704
• https://doi.org/10.4213/im704
• http://mi.mathnet.ru/eng/izv/v70/i3/p129

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This publication is cited in the following articles:
1. V. V. Lebedev, “On $l^p$-Multipliers of Functions Analytic in the Disk”, Funct. Anal. Appl., 48:3 (2014), 231–234
2. Lebedev V., “Thickness Conditions and Littlewood-Paley Sets”, Studia Math., 220:3 (2014), 265–276
3. Cheng R., Mashreghi J., Ross W.T., “Multipliers of Sequence Spaces”, Concr. Operators, 4:1 (2017), 76–108
4. Lebedev V., “Sets With Distinct Sums of Pairs, Long Arithmetic Progressions, and Continuous Mappings”, Anal. Math., 44:3 (2018), 369–380
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