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