  Let $\Omega\in L^1{({\mathbb S^{n-1}})}$, be a function of homogeneous of
degree zero, and $M_\Omega$ be the Hardy-Littlewood maximal operator associated
with $\Omega$ defined by $M_\Omega(f)(x) =
\sup_{r>0}\frac1{r^n}\int_{|x-y|<r}|\Omega(x-y)f(y)|dy.$ It was shown by Christ
and Rubio de Francia that $\|M_\Omega(f)\|_{L^{1,\infty}({\mathbb R^n})} \le
C(\|\Omega\|_{L\log L({\mathbb S^{n-1}})}+1)\|f\|_{L^1({\mathbb R^n})}$
provided $\Omega\in L\log L {({\mathbb S^{n-1}})}$. In this paper, we show
that, if $\Omega\in L\log L({\mathbb S^{n-1}})$, then for all $f\in
L^1({\mathbb R^n})$, $M_\Omega$ enjoys the limiting weak-type behaviors that
$$\lim_{\lambda\to 0^+}\lambda|\{x\in{\mathbb R^n}:M_\Omega(f)(x)>\lambda\}| =
n^{-1}\|\Omega\|_{L^1({\mathbb S^{n-1}})}\|f\|_{L^1({\mathbb R^n})}.$$ This
removes the smoothness restrictions on the kernel $\Omega$, such as Dini-type
conditions, in previous results. To prove our result, we present a new upper
bound of $\|M_\Omega\|_{L^1\to L^{1,\infty}}$, which essentially improves the
upper bound $C(\|\Omega\|_{L\log L({\mathbb S^{n-1}})}+1)$ given by Christ and
Rubio de Francia. As a consequence, the upper and lower bounds of
$\|M_\Omega\|_{L^1\to L^{1,\infty}}$ are obtained for $\Omega\in L\log L
{({\mathbb S^{n-1}})}$.
