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VARIANTS OF A MULTIPLIER THEOREM OF KISLYAKOV

Part of: Harmonic analysis in one variable Abstract harmonic analysis

Published online by Cambridge University Press:  01 September 2022

Andreas Defant ,
Mieczysław Mastyło Open the ORCID record for Mieczysław Mastyło [Opens in a new window]  and
Antonio Pérez-Hernández Open the ORCID record for Antonio Pérez-Hernández [Opens in a new window]
Andreas Defant
Affiliation:
Institut für Mathematik, Carl von Ossietzky Universität, Postfach 2503 D-26111 Oldenburg, Germany (andreas.defant@uni-oldenburg.de)
Mieczysław Mastyło
Affiliation:
Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Poznań, Uniwersytetu Poznańskiego 4, 61-614 Poznań, Poland (mieczyslaw.mastylo@amu.edu.pl)
Antonio Pérez-Hernández*
Affiliation:
Departamento de Matemática Aplicada I, Escuela Técnica Superior de Ingenieros Industriales, Universidad Nacional de Educación a Distancia (UNED), 28040 Madrid, Spain
*
antperez@ind.uned.es
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Abstract

We prove stronger variants of a multiplier theorem of Kislyakov. The key ingredients are based on ideas of Kislyakov and the Kahane–Salem–Zygmund inequality. As a by-product, we show various multiplier theorems for spaces of trigonometric polynomials on the n-dimensional torus $\mathbb {T}^n$ or Boolean cubes $\{-1,1\}^N$. Our more abstract approach based on local Banach space theory has the advantage that it allows to consider more general compact abelian groups instead of only the multidimensional torus. As an application, we show that various recent $\ell _1$-multiplier theorems for trigonometric polynomials in several variables or ordinary Dirichlet series may be proved without the Kahane–Salem–Zygmund inequality.

Keywords

Fourier analysis on groupsmultipliersSidon setsBoolean functionsmultivariable and Dirichlet polynomials

MSC classification

Primary: 43A46: Special sets (thin sets, Kronecker sets, Helson sets, Ditkin sets, Sidon sets, etc.) 43A75: Analysis on specific compact groups 06E30: Boolean functions 42A16: Fourier coefficients, Fourier series of functions with special properties, special Fourier series
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 23 , Issue 1 , January 2024 , pp. 347 - 377
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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