{
  "id": "2402.09787",
  "version": "v1",
  "published": "2024-02-15T08:35:35.000Z",
  "updated": "2024-02-15T08:35:35.000Z",
  "title": "Critical exponents of the Riesz projection",
  "authors": [
    "Ole Fredrik Brevig",
    "Adrián Llinares",
    "Kristian Seip"
  ],
  "doi": "10.1007/s40315-024-00566-z",
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $\\mathfrak{p}_d(q)$ denote the critical exponent of the Riesz projection from $L^q(\\mathbb{T}^d)$ to the Hardy space $H^p(\\mathbb{T}^d)$, where $\\mathbb{T}$ is the unit circle. We present the state-of-the-art on the conjecture that $\\mathfrak{p}_1(q) = 4(1-1/q)$ for $1 \\leq q \\leq \\infty$ and prove that it holds in the endpoint case $q = 1$. We then extend the conjecture to \\[\\mathfrak{p}_d(q) = 2+\\cfrac{2}{d+\\cfrac{2}{q-2}}\\] for $d\\geq1$ and $\\frac{2d}{d+1} \\leq q \\leq \\infty$ and establish that if the conjecture holds for $d=1$, then it also holds for $d=2$. When $d=2$, we verify that the conjecture holds in the endpoint case $q = 4/3$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-02-15T08:35:35.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "riesz projection",
      "critical exponent",
      "conjecture holds",
      "endpoint case",
      "hardy space"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Springer"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}