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.
@article{1049123078,
author = {Carbery, Anthony and Ricci, Fulvio and Wright, James},
title = {Maximal functions and singular integrals associated to
polynomial mappings of $\mathbb{R}^n$},
journal = {Rev. Mat. Iberoamericana},
volume = {19},
number = {2},
year = {2003},
pages = { 1-22},
language = {en},
url = {http://dml.mathdoc.fr/item/1049123078}
}
Carbery, Anthony; Ricci, Fulvio; Wright, James. Maximal functions and singular integrals associated to
polynomial mappings of $\mathbb{R}^n$. Rev. Mat. Iberoamericana, Tome 19 (2003) no. 2, pp. 1-22. http://gdmltest.u-ga.fr/item/1049123078/