Singular integral operators on function spaces

We study the singular integral operator fx, t(y′) = f (x - ty′), defined on all test functions f, where b is a bounded function, α ≥ 0, Ω is suitable distribution on the unit sphere S ^n-1 satisfying some cancellation conditions. We prove certain boundedness properties of T _Ω,α on the Triebel-Lizorkin spaces and on the Besov spaces. We also use our results to study the Littlewood-Paley functions. These results improve and extend some well-known results.