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.