{
  "id": "2203.15303",
  "version": "v1",
  "published": "2022-03-29T07:45:36.000Z",
  "updated": "2022-03-29T07:45:36.000Z",
  "title": "Pseudodifferential operators on Mixed-Norm $α$-modulation spaces",
  "authors": [
    "Morten Nielsen"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "Mixed-norm $\\alpha$-modulation spaces were introduced recently by Cleanthous and Georgiadis [Trans.\\ Amer.\\ Math.\\ Soc.\\ 373 (2020), no. 5, 3323-3356]. The mixed-norm spaces $M^{s,\\alpha}_{\\vec{p},q}(\\mathbb{R}^n)$, $\\alpha\\in [0,1]$, form a family of smoothness spaces that contain the mixed-norm Besov spaces as special cases. In this paper we prove that a pseudodifferential operator $\\sigma(x,D)$ with symbol in the H\\\"ormander class $S^b_{\\rho}$ extends to a bounded operator $\\sigma(x,D)\\colon M^{s,\\alpha}_{\\vec{p},q}(\\mathbb{R}^n) \\rightarrow M^{s-b,\\alpha}_{\\vec{p},q}(\\mathbb{R}^n)$ provided $0<\\alpha\\leq \\rho\\leq 1$, $\\vec{p}\\in (0,\\infty)^n$, and $0<q<\\infty$. The result extends the known result that pseudodifferential operators with symbol in the class $S^b_{1}$ maps the mixed-norm Besov space $B^s_{\\vec{p},q}(\\mathbb{R}^n)$ into $B^{s-b}_{\\vec{p},q}(\\mathbb{R}^n)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-29T07:45:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "47G30",
      "46E35",
      "47B38"
    ],
    "keywords": [
      "pseudodifferential operator",
      "modulation spaces",
      "mixed-norm besov space",
      "special cases",
      "mixed-norm spaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}