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 ${\mathbb{S}}^{n-1}(n\ge 2)$. 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.

21 pages