{
  "id": "1307.5642",
  "version": "v2",
  "published": "2013-07-22T10:10:56.000Z",
  "updated": "2013-11-29T09:36:17.000Z",
  "title": "Optimal exponents in weighted estimates without examples",
  "authors": [
    "Teresa Luque",
    "Carlos Pérez",
    "Ezequiel Rela"
  ],
  "comment": "Revised and corrected version. To appear in Math. Res. Lett",
  "categories": [
    "math.CA"
  ],
  "abstract": "We present a general approach for proving the optimality of the exponents on weighted estimates. We show that if an operator $T$ satisfies a bound like $$ \\|T\\|_{L^{p}(w)}\\le c\\, [w]^{\\beta}_{A_p} \\qquad w \\in A_{p}, $$ then the optimal lower bound for $\\beta$ is closely related to the asymptotic behaviour of the unweighted $L^p$ norm $\\|T\\|_{L^p(\\mathbb{R}^n)}$ as $p$ goes to 1 and $+\\infty$, which is related to Yano's classical extrapolation theorem. By combining these results with the known weighted inequalities, we derive the sharpness of the exponents, without building any specific example, for a wide class of operators including maximal-type, Calder\\'on--Zygmund and fractional operators. In particular, we obtain a lower bound for the best possible exponent for Bochner-Riesz multipliers. We also present a new result concerning a continuum family of maximal operators on the scale of logarithmic Orlicz functions. Further, our method allows to consider in a unified way maximal operators defined over very general Muckenhoupt bases.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-11-29T09:36:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "weighted estimates",
      "optimal exponents",
      "logarithmic orlicz functions",
      "unified way maximal operators"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1307.5642L"
    }
  }
}