We study the singular integral operator
$${{T}_{\Omega ,\alpha }}f\left( x \right)\,=\,\text{p}\text{.v}\text{.}\,{{\int }_{{{R}^{n}}}}\,b\left( \left| y \right| \right)\Omega \left( {{y}'} \right){{\left| y \right|}^{-n-\alpha }}\,f\left( x\,-\,y \right)\,dy,$$
defined on all test functions
$f$
, where
$b$
is a bounded function,
$\alpha \ge 0,\,\Omega \left( {{y}'} \right)$
is an integrable function on the unit sphere
${{S}^{n-1}}$
satisfying certain cancellation conditions. We prove that, for
$1\,<\,p\,<\infty$
,
${{T}_{\Omega ,\alpha }}$
extends to a bounded operator from the Sobolev space
$L_{\alpha }^{p}$
to the Lebesgue space
${{L}^{p}}$
with
$\Omega$
being a distribution in the Hardy space
${{H}^{q}}\left( {{S}^{n-1}} \right)$
where
$q=\frac{n-1}{n-1+\alpha }$
. The result extends some known results on the singular integral operators. As applications, we obtain the boundedness for
${{T}_{\Omega ,\alpha }}$
on the Hardy spaces, as well as the boundedness for the truncated maximal operator
$T_{\Omega ,m}^{*}$
.