{
  "id": "2203.04360",
  "version": "v1",
  "published": "2022-03-08T19:34:14.000Z",
  "updated": "2022-03-08T19:34:14.000Z",
  "title": "Mixed inequalities of Fefferman-Stein type for singular integral operators",
  "authors": [
    "Fabio Berra",
    "Marilina Carena",
    "Gladis Pradolini"
  ],
  "comment": "17 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "We give Feffermain-Stein type inequalities related to mixed estimates for Calder\\'on-Zygmund operators. More precisely, given $\\delta>0$, $q>1$, $\\varphi(z)=z(1+\\log^+z)^\\delta$, a nonnegative and locally integrable function $u$ and $v\\in \\mathrm{RH}_\\infty\\cap A_q$, we prove that the inequality \\[uv\\left(\\left\\{x\\in \\mathbb{R}^n: \\frac{|T(fv)(x)|}{v(x)}>t\\right\\}\\right)\\leq \\frac{C}{t}\\int_{\\mathbb{R}^n}|f|\\left(M_{\\varphi, v^{1-q'}}u\\right)M(\\Psi(v))\\] holds with $\\Psi(z)=z^{p'+1-q'}\\mathcal{X}_{[0,1]}(z)+z^{p'}\\mathcal{X}_{[1,\\infty)}(z)$, for every $t>0$ and every $p>\\max\\{q,1+1/\\delta\\}$. This inequality provides a more general version of mixed estimates for Calder\\'on-Zygmund operators proved in \\cite{CruzUribe-Martell-Perez}. It also generalizes the Fefferman-Stein estimates given in \\cite{P94} for the same operators. We further get similar estimates for operators of convolution type with kernels satisfying an $L^\\Phi-$H\\\"ormander condition, generalizing some previously known results which involve mixed estimates and Fefferman-Stein inequalities for these operators.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-08T19:34:14.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "singular integral operators",
      "fefferman-stein type",
      "inequality",
      "mixed inequalities",
      "mixed estimates"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}