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/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. 
Nikolai K. Nikolski and Igor E. Verbitsky
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