A singular integral operator with rough kernel

Abstract:

Let $b(y)$ be a bounded radial function and $\Omega (y’)$ an $H^1$ function on the unit sphere satisfying the mean zero property. Under certain growth conditions on $\Phi (t)$, we prove that the singular integral operator \begin{equation*} T_{\Phi ,b}f(x)=\text {p.v.} \int _{\mathbb R^n} f(x-\Phi (|y|)y’) b(y)|y|^{-n}\Omega (y’) dy \end{equation*} is bounded in $L^p(\mathbb R^n)$ for $1<p<\infty$.