Variants of a multiplier theorem of Kislyakov

TL;DR

This paper enhances a multiplier theorem of Kislyakov using advanced techniques, broadening its applicability to various function spaces and groups, and simplifies proofs of recent results in harmonic analysis.

Contribution

It provides stronger variants of Kislyakov's multiplier theorem and extends the scope to general compact abelian groups using local Banach space theory.

Findings

Stronger multiplier theorems for trigonometric polynomials and Boolean cubes.

Extension of results to general compact abelian groups.

Simplified proofs of recent $ ext{ell}_1$-multiplier theorems without Kahane-Salem-Zygmund inequality.

Abstract

We prove stronger variants of a multiplier theorem of Kislyakov. The key ingredients are based on ideas of Kislaykov 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.