Nikolai K. Nikolski, Igor E. Verbitsky
Published 2014-04-16, updated 2015-09-28Version 2
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.
Comments: Version 2: Remark at the end of Sec. 2 deleted, discussion of the splitting case added (Remarks after Lemma 2.1, Lemma 2.3, Lemma 3.8), Theorem 3.4 modified in the case $1/w \not\in L^1(\mathbb{T})$, Secs. 4-5 unchanged
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