Variants of a multiplier theorem of Kislyakov

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.