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

Abstract

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