{
  "id": "1404.4380",
  "version": "v2",
  "published": "2014-04-16T20:27:59.000Z",
  "updated": "2015-09-28T16:33:57.000Z",
  "title": "Fourier multipliers for weighted $L^{2}$ spaces with Lévy-Khinchin-Schoenberg weights",
  "authors": [
    "Nikolai K. Nikolski",
    "Igor E. Verbitsky"
  ],
  "comment": "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",
  "categories": [
    "math.FA",
    "math.CA",
    "math.SP"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-04-16T20:27:59.000Z",
      "abstract": "We present a class of weight functions $ w$ on the circle $ \\mathbb{T}$, called L\\'evy-Khinchin-Schoenberg ({\\rm 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 {\\rm LKS} requirements. Some more complicated compositions of {\\rm LKS} weights are considered as well.",
      "comment": "47 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-09-28T16:33:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B15",
      "46B15",
      "47A10"
    ],
    "keywords": [
      "fourier multipliers",
      "lévy-khinchin-schoenberg weights",
      "spectral localization property",
      "case multipliers satisfy",
      "capacitary inequality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 47,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1404.4380N"
    }
  }
}