Rough maximal functions and rough singular integral operators applied to integrable radial functions

Abstract

Let $\Omega$ be homogeneous of degree 0 in ${\mathbb{R}}^n$ and integrable on the unit sphere. A rough maximal operator is obtained by inserting a factor $\Omega$ in the denition of the ordinary maximal function. Rough singular integral operators are given by principal value kernels $\Omega (y)/|y{|}^n$, provided that the mean value of $\Omega$ vanishes. In an earlier paper, the authors showed that a two-dimensional rough maximal operator is of weak type (1,1) when restricted to radial functions. This result is now extended to arbitrary finite dimension, and to rough singular integrals.