Composition of rough singular integral operators on rearrangement invariant Banach type spaces

Let $\mathrm{\Omega }$ be a homogeneous function of degree zero and enjoy the vanishing condition on the unit sphere ${\mathbb{S}}^{n-1}(n\ge 2)$. Let $T_{\mathrm{\Omega }}$ be the convolution singular integral operator with kernel $\mathrm{\Omega }(x)\mathrm{\mid }x{\mathrm{\mid }}^{-n}$. In this paper, when $\mathrm{\Omega }\in L^{\mathrm{\infty }}({\mathbb{S}}^{n-1})$, we consider the quantitative weighted bounds of the composite operators of $T_{\mathrm{\Omega }}$ 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.