# Revista Matemática Iberoamericana

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**Volume 19, Issue 1, 2003, pp. 1–22**

**DOI: 10.4171/RMI/336**

Published online: 2003-04-30

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

Anthony Carbery^{[1]}, Fulvio Ricci

^{[2]}and James Wright

^{[3]}(1) University of Edinburgh, UK

(2) Scuola Normale Superiore, Pisa, Italy

(3) University of Edinburgh, UK

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.

*Keywords: *Maximal functions, singular integrals, weak-type estimates

Carbery Anthony, Ricci Fulvio, Wright James: Maximal functions and singular integrals associated to polynomial mappings of $\mathbb{R}^n$. *Rev. Mat. Iberoamericana* 19 (2003), 1-22. doi: 10.4171/RMI/336