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

Abstract

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

, 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.