{
  "id": "2302.11877",
  "version": "v1",
  "published": "2023-02-23T09:28:11.000Z",
  "updated": "2023-02-23T09:28:11.000Z",
  "title": "Some sharp inequalities of Mizohata--Takeuchi-type",
  "authors": [
    "Anthony Carbery",
    "Marina Iliopoulou",
    "Hong Wang"
  ],
  "categories": [
    "math.CA",
    "math.AP"
  ],
  "abstract": "Let $\\Sigma$ be a strictly convex, compact patch of a $C^2$ hypersurface in $\\mathbb{R}^n$, with non-vanishing Gaussian curvature and surface measure $d\\sigma$ induced by the Lebesgue measure in $\\mathbb{R}^n$. The Mizohata--Takeuchi conjecture states that \\begin{equation*} \\int |\\widehat{gd\\sigma}|^2w \\leq C \\|Xw\\|_\\infty \\int |g|^2 \\end{equation*} for all $g\\in L^2(\\Sigma)$ and all weights $w:\\mathbb{R}^n\\rightarrow [0,+\\infty)$, where $X$ denotes the $X$-ray transform. As partial progress towards the conjecture, we show, as a straightforward consequence of recently-established decoupling inequalities, that for every $\\epsilon>0$, there exists a positive constant $C_\\epsilon$, which depends only on $\\Sigma$ and $\\epsilon$, such that for all $R \\geq 1$ and all weights $w:\\mathbb{R}^n\\rightarrow [0,+\\infty)$ we have \\begin{equation*} \\int_{B_R} |\\widehat{gd\\sigma}|^2w \\leq C_\\epsilon R^\\epsilon \\sup_T \\left(\\int _T w^{\\frac{n+1}{2}}\\right)^{\\frac{2}{n+1}}\\int |g|^2, \\end{equation*} where $T$ ranges over the family of all tubes in $\\mathbb{R}^n$ of dimensions $R^{1/2} \\times \\dots \\times R^{1/2} \\times R$. From this we deduce the Mizohata--Takeuchi conjecture with an $R^{\\frac{n-1}{n+1}}$-loss; i.e., that \\begin{equation*} \\int_{B_R} |\\widehat{gd\\sigma}|^2w \\leq C_\\epsilon R^{\\frac{n-1}{n+1}+ \\epsilon}\\|Xw\\|_\\infty\\int |g|^2 \\end{equation*} for any ball $B_R$ of radius $R$ and any $\\epsilon>0$. The power $(n-1)/(n+1)$ here cannot be replaced by anything smaller unless properties of $\\widehat{gd\\sigma}$ beyond 'decoupling axioms' are exploited. We also provide estimates which improve this inequality under various conditions on the weight, and discuss some new cases where the conjecture holds.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-02-23T09:28:11.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "sharp inequalities",
      "inequality",
      "mizohata-takeuchi-type",
      "mizohata-takeuchi conjecture states",
      "surface measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}