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

### Anthony Carbery

University of Edinburgh, UK### Fulvio Ricci

Scuola Normale Superiore, Pisa, Italy### James Wright

University of Edinburgh, UK

## Abstract

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

$T_Pf(x) =\int_{\mathbb{R}^m} f\big(x-P(y)\big)K(y) dy$

, where $P$ is a polynomial defined on $\mathbb{R}^m$ with values in $\mathbb{R}^n$ and $K$ is a smooth Calderón-Zygmund kernel on $\mathbb{R}^m$. A maximal operator $M_P$ can be constructed in a similar fashion. We discuss weak-type 1-1 estimates for $T_P$ and $M_P$ and the uniformity of such estimates with respect to $P$. We also obtain $L^p$-estimates for "supermaximal" operators, defined by taking suprema over $P$ 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 $\mathbb{R}^n$. Rev. Mat. Iberoam. 19 (2003), no. 1, pp. 1–22

DOI 10.4171/RMI/336