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 Sn−1 satisfying some cancellation conditions. We prove certain boundedness properties of TΩ,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.  2002 Elsevier Science (USA). All rights reserved.