JournalsrmiVol. 19, No. 1pp. 1–22

Maximal functions and singular integrals associated to polynomial mappings of Rn\mathbb{R}^n

  • Anthony Carbery

    University of Edinburgh, UK
  • Fulvio Ricci

    Scuola Normale Superiore, Pisa, Italy
  • James Wright

    University of Edinburgh, UK
Maximal functions and singular integrals associated to polynomial mappings of $\mathbb{R}^n$ cover
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Abstract

We consider convolution operators on Rn\mathbb{R}^n of the form

TPf(x)=Rmf(xP(y))K(y)dyT_Pf(x) =\int_{\mathbb{R}^m} f\big(x-P(y)\big)K(y) dy

, where PP is a polynomial defined on Rm\mathbb{R}^m with values in Rn\mathbb{R}^n and KK is a smooth Calderón-Zygmund kernel on Rm\mathbb{R}^m. A maximal operator MPM_P can be constructed in a similar fashion. We discuss weak-type 1-1 estimates for TPT_P and MPM_P and the uniformity of such estimates with respect to PP. We also obtain LpL^p-estimates for "supermaximal" operators, defined by taking suprema over PP ranging in certain classes of polynomials of bounded degree.

Cite this article

Anthony Carbery, Fulvio Ricci, James Wright, Maximal functions and singular integrals associated to polynomial mappings of Rn\mathbb{R}^n. Rev. Mat. Iberoam. 19 (2003), no. 1, pp. 1–22

DOI 10.4171/RMI/336