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

Abstract

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.