{
  "id": "1506.07005",
  "version": "v1",
  "published": "2015-06-14T15:39:18.000Z",
  "updated": "2015-06-14T15:39:18.000Z",
  "title": "$L^p$-Asymptotics of Fourier transform of fractal measures",
  "authors": [
    "K. S. Senthil Raani"
  ],
  "comment": "Thesis submitted in Indian Institute of Science, Bangalore",
  "categories": [
    "math.CA",
    "math.FA"
  ],
  "abstract": "One of the basic questions in harmonic analysis is to study the decay properties of the Fourier transform of measures or distributions supported on thin sets in $\\mathbb{R}^n$. When the support is a smooth enough manifold, an almost complete picture is available. One of the early results in this direction is the following: Let $f\\in C_c^{\\infty}(\\mathbb{R}^n)$ and $d\\sigma$ be the surface measure on the sphere $S^{n-1}\\subset\\mathbb{R}^n$. Then $$|\\widehat{fd\\sigma}(\\xi)|\\leq\\ C\\ (1+|\\xi|)^{-\\frac{n-1}{2}}.$$ It follows that $\\widehat{fd\\sigma}\\in L^p(\\mathbb{R}^n)$ for all $p>\\frac{2n}{n-1}$. This result can be extended to compactly supported measure on $(n-1)$-dimensional manifolds with appropriate assumptions on the curvature. Similar results are known for measures supported in lower dimensional manifolds in $\\mathbb{R}^n$ under appropriate curvature conditions. However, the picture for fractal measures is far from complete. This thesis is a contribution to the study of asymptotic properties of the Fourier transform of measures supported in sets of fractal dimension $0<\\alpha<n$ for $p\\leq 2n/\\alpha$. In 2004, Agranovsky and Narayanan proved that if $\\mu$ is a measure supported in a $C^1$-manifold of dimension $d<n$, then $\\widehat{fd\\mu}\\notin L^p(\\mathbb{R}^n)$ for $1\\leq p\\leq \\frac{2n}{d}$. We prove that the Fourier transform of a measure $\\mu_E$ supported in a set $E$ of fractal dimension $\\alpha$ does not belong to $L^p(\\mathbb{R}^n)$ for $p\\leq 2n/\\alpha$. We also study $L^p$-asymptotics of the Fourier transform of fractal measures $\\mu_E$ under appropriate conditions on $E$ and give quantitative versions of the above statement by obtaining lower and upper bounds for the following: $$\\underset{L\\rightarrow\\infty}{\\limsup} \\frac{1}{L^k} \\int_{|\\xi|\\leq L}|\\widehat{fd\\mu_E}(\\xi)|^pd\\xi.$$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-06-14T15:39:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B25",
      "42B10",
      "28A78",
      "28A80",
      "26D15",
      "37F35"
    ],
    "keywords": [
      "fourier transform",
      "fractal measures",
      "asymptotic",
      "fractal dimension",
      "appropriate curvature conditions"
    ],
    "tags": [
      "dissertation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}