Fourier multipliers for weighted $L^2$ spaces with L\'evy-Khinchin-Schoenberg weights

We present a class of weight functions $w$ on the circle $\mathbb{T}$, called L'evy-Khinchin-Schoenberg (LKS) weights, for which we are able to completely characterize (in terms of a capacitary inequality) all Fourier multipliers for the weighted space $L^2(\mathbb{T},w)$. We show that the multiplier algebra is nontrivial if and only if $1\mathrm{/}w\in L^1(\mathbb{T})$, and in this case multipliers satisfy the Spectral Localization Property (no “hidden spectrum”). On the other hand, the Muckenhoupt $(A_2)$ condition responsible for the basis property of exponentials $(e^{ikx})$ is more or less independent of the Spectral Localization Property and LKS requirements. Some more complicated compositions of LKS weights are considered as well.