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arXiv · 1404.4380

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

Nikolai K. Nikolski·Igor E. Verbitsky
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Abstract

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.

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BibTeXRIS

Nikolai K. Nikolski, Igor E. Verbitsky. 2014-04-16. Fourier multipliers for weighted $L^{2}$ spaces with L\'evy-Khinchin-Schoenberg weights. https://doi.org/10.1515/crelle-2014-0123

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