  Let $\Omega$ be a homogeneous function of degree zero and enjoy the vanishing
condition on the unit sphere $\mathbb{S}^{n-1}(n\geq 2)$. Let $T_{\Omega}$ be
the convolution singular integral operator with kernel ${\Omega(x)}{|x|^{-n}}$.
In this paper, when $\Omega \in L^{\infty}(\mathbb {S}^{n-1})$, we consider the
quantitative weighted bounds of the composite operators of $T_{\Omega}$ on
rearrangement invariant Banach function spaces. These spaces contain the
classical Lorentz spaces and Orlicz spaces as special examples. Weighted
boundedness of the composite operators on rearrangement invariant quasi-Banach
spaces were also given.
