{
  "id": "2308.03912",
  "version": "v1",
  "published": "2023-08-07T21:17:39.000Z",
  "updated": "2023-08-07T21:17:39.000Z",
  "title": "Convolution Operators in Matrix Weighted, Variable Lebesgue Spaces",
  "authors": [
    "David Cruz-Uribe",
    "Michael Penrod"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "We extend the theory of matrix weights to the variable Lebesgue spaces. The theory of matrix $\\mathcal{A}_p$ weights has attracted considerable attention beginning with the work of Nazarov, Treil, and Volberg in the 1990s. We extend this theory by generalizing the matrix $\\mathcal{A}_p$ condition to the variable exponent setting. We prove boundedness of the convolution operator $\\mathbf{f}\\mapsto \\phi\\ast \\mathbf{F}$ for $\\phi \\in C_c^\\infty(\\Omega)$, and show that the approximate identity defined using $\\phi$ converges in matrix weighted, variable Lebesgue spaces $L^{p(\\cdot)}(W,\\Omega)$ for $W$ in matrix $\\mathcal{A}_{p(\\cdot)}$. Our approach to this problem is through averaging operators; these results are of interest in their own right. As an application of our work, we prove a version of the classical $H=W$ theorem for matrix weighted, variable exponent Sobolev spaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-08-07T21:17:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B25",
      "42B35",
      "46E35"
    ],
    "keywords": [
      "variable lebesgue spaces",
      "convolution operator",
      "variable exponent sobolev spaces",
      "approximate identity",
      "matrix weights"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}