  In this paper, we establish some bilinear endpoint estimates of Calder\'on
commutator $\mathcal{C}[\nabla A,f](x)$ with a homogeneous kernel when
$\Omega\in L\log^+L(\mathbf{S}^{d-1})$. More precisely, we prove that
$\mathcal{C}[\nabla A,f]$ maps $L^q(\mathbb{R}^d)\times L^1(\mathbb{R}^d)$ to
$L^{r,\infty}(\mathbb{R}^d)$ if $q>d$ which improves previous result
essentially. If $q=d$, we show that Calder\'on commutator maps
$L^{d,1}(\mathbb{R}^d)\times L^1(\mathbb{R}^d)$ to $L^{r,\infty}(\mathbb{R}^d)$
which is new even if the kernel is smooth. The novelty in the paper is that we
prove a new endpoint estimate of the Mary Weiss maximal function which may have
its own interest in the theory of singular integral.
