  In 2006, Janakiraman [10] showed that if $\Omega$ with mean value zero on
$S^{n-1}$ satisfies the condition \[
\sup_{|\xi|=1}\int_{S^{n-1}}|\Omega(\theta)-\Omega(\theta+\delta\xi)|d\sigma(\theta)\leq
Cn\delta\int_{S^{n-1}}|\Omega(\theta)|d\sigma(\theta),\quad
0<\delta<\frac{1}{n},\ (\ast) \] then for the singular integral operator
$T_\Omega$ with homogeneous kernel, the following limiting behavior holds:
\[\lim\limits_{\lambda\rightarrow 0}\lambda m(\{x\in\mathbb{R}^n:|T_\Omega
f(x)|>\lambda\})= \frac{1}{n}\|\Omega\|_{1}\|f\|_{1},\quad \text{for}\ f\in
L^1(\mathbb{R}^n)\ \text{with}\ f\geq 0.\ (\ast\ast)\]
  In the present paper, we prove that if replacing the condition $(\ast)$ by
more general condition, the $L^1$-Dini condition, then the limiting behavior
$(\ast\ast)$ still holds for the singular integral $T_\Omega$. In particular,
we give an example which satisfies the $L^1$-Dini condition, but does not
satisfy $(\ast)$. Hence, we improve essentially the above result given in [10].
To prove our conclusion, we show that the $L^1$-Dini conditions defined
respectively via the rotation and translation on $\mathbb{R}^n$ are equivalent
(see Theorem 2.5 below), which has its own interest in the theory of singular
integrals. Moreover, similar limiting behavior for the fractional integral
operator $T_{\Omega,\alpha}$ with homogeneous kernel is also established in
this paper.
