{
  "id": "2211.13646",
  "version": "v1",
  "published": "2022-11-24T14:51:22.000Z",
  "updated": "2022-11-24T14:51:22.000Z",
  "title": "Singular integrals along variable codimension one subspaces",
  "authors": [
    "Odysseas Bakas",
    "Francesco Di Plinio",
    "Ioannis Parissis",
    "Luz Roncal"
  ],
  "comment": "49 pages, 3 figures. Submitted for publication",
  "categories": [
    "math.CA"
  ],
  "abstract": "This article deals with maximal operators on ${\\mathbb R}^n$ formed by taking arbitrary rotations of tensor products of a $d$-dimensional H\\\"ormander-Mihlin multiplier with the identity in $n-d$ coordinates, in the particular codimension 1 case $d=n-1$. These maximal operators are naturally connected to differentiation problems and maximally modulated singular integrals such as Sj\\\"olin's generalization of Carleson's maximal operator. Our main result, a weak-type $L^{2}({\\mathbb R}^n)$-estimate on band-limited functions, leads to several corollaries. The first is a sharp $L^2({\\mathbb R}^n)$ estimate for the maximal operator restricted to a finite set of rotations in terms of the cardinality of the finite set. The second is a version of the Carleson-Sj\\\"olin theorem. In addition, we obtain that functions in the Besov space $B_{p,1}^0({\\mathbb R}^n)$, $2\\le p <\\infty$, may be recovered from their averages along a measurable choice of codimension $1$ subspaces, a form of Zygmund's conjecture in general dimension $n$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-24T14:51:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B20"
    ],
    "keywords": [
      "variable codimension",
      "finite set",
      "carlesons maximal operator",
      "tensor products",
      "differentiation problems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 49,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}