  Convolution type Calder\'on-Zygmund singular integral operators with rough
kernels $\pv \Om(x)/|x|^n$ are studied. A condition on $\Om$ implying that the
corresponding singular integrals and maximal singular integrals map $L^p \to
L^p$ for $1<p<\nf$ is obtained. This condition is shown to be different from
the condition $\Om\in H^1(\sn)$.
