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multivariable and dirichlet polynomials

Journal of the Institute of Mathematics of Jussieu (1)

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VARIANTS OF A MULTIPLIER THEOREM OF KISLYAKOV
Part of
- Harmonic analysis in one variable
- Abstract harmonic analysis
Andreas Defant, Mieczysław Mastyło, Antonio Pérez-Hernández
Journal:

Journal of the Institute of Mathematics of Jussieu / Volume 23 / Issue 1 / January 2024

Published online by Cambridge University Press:

01 September 2022, pp. 347-377

Print publication:

January 2024
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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.