Rough singular integrals on Triebel–Lizorkin space and Besov space

In this paper the authors prove that the homogeneous singular integral TΩ with Ω∈H1(Sn−1) is bounded on the Triebel–Lizorkin spaces and the Besov spaces. These results answer an open problem proposed by Chen and Zhang in [J. Chen, C. Zhang, Boundedness of rough singular integral on the Triebel–Lizorkin spaces, J. Math. Anal. Appl. 337 (2008) 1048–1052]. The same results hold also for the rough singular integral operators TΩ,h with radial function kernels.