{
  "id": "2407.19982",
  "version": "v1",
  "published": "2024-07-29T13:17:24.000Z",
  "updated": "2024-07-29T13:17:24.000Z",
  "title": "On inversion of absolutely convergent weighted Dirichlet series in two variables",
  "authors": [
    "Prakash A. Dabhi"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $0<p\\leq 1$, and let $\\omega:\\mathbb N^2 \\to [1,\\infty)$ be an almost monotone weight. Let $\\mathbb H$ be the closed right half plane in the complex plane. Let $\\widetilde a$ be a complex valued function on $\\mathbb H^2$ such that $\\widetilde a(s_1,s_2)=\\sum_{(m,n)\\in \\mathbb N^2}a(m,n)m^{-s_1}n^{-s_2}$ for all $(s_1,s_2)\\in \\mathbb H^2$ with $\\sum_{(m,n)\\in \\mathbb N^2} |a(m,n)|^p\\omega(m,n)<\\infty$. If $\\widetilde a$ is bounded away from zero on $\\mathbb H^2$, then there is an almost monotone weight $\\nu$ on $\\mathbb N^2$ such that $1\\leq \\nu\\leq \\omega$, $\\nu$ is constant if and only if $\\omega$ is constant, $\\nu$ is admissible if and only if $\\omega$ is admissible, the reciprocal $\\frac{1}{\\widetilde a}$ has the Dirichlet representation $\\frac{1}{\\widetilde a}(s_1,s_2)=\\sum_{(m,n)\\in \\mathbb N^2}b(m,n)m^{-s_1}n^{-s_2}$ for all $(s_1,s_2)\\in \\mathbb H^2$ and $\\sum_{(m,n)\\in \\mathbb N^2}|b(m,n)|^p\\nu(m,n)<\\infty$. If $\\varphi$ is holomorphic on a neighbourhood of the closure of range of $\\widetilde a$, then there is an almost monotone weight $\\xi$ on $\\mathbb N^2$ such that $1\\leq \\xi\\leq \\omega$, $\\xi$ is constant if and only if $\\omega$ is constant, $\\xi$ is admissible if and only if $\\omega$ is admissible, $\\varphi \\circ \\widetilde a$ has the Dirichlet series representation $(\\varphi\\circ \\widetilde a)(s_1,s_2)=\\sum_{(m,n)\\in \\mathbb N^2} c(m,n)m^{-s_1}n^{-s_2}\\;((s_1,s_2)\\in \\mathbb H^2)$ and $\\sum_{(m,n)\\in \\mathbb N^2}|c(m,n)|^p\\xi(m,n)<\\infty$. Let $\\omega$ be an admissible weight on $\\mathbb N^2$, and let $\\widetilde a$ have $p$-th power $\\omega$- absolutely convergent Dirichlet series. Then it is shown that the reciprocal of $\\widetilde a$ has $p$-th power $\\omega$- absolutely convergent Dirichlet series if and only if $\\widetilde a$ is bounded away from zero.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-29T13:17:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "30B50",
      "30B50",
      "46H99",
      "46J05"
    ],
    "keywords": [
      "absolutely convergent weighted dirichlet series",
      "absolutely convergent dirichlet series",
      "monotone weight",
      "th power",
      "right half plane"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}