A note on weak type L1L1 boundedness of CZOs with rough kernels

For f∈S(R2)f∈S(R2) and Ω∈L1(S1)Ω∈L1(S1), ∫S1Ω(x′)dx′=0, defineTΩ(f)(x)=limϵ→0+∫|x−y|⩾ϵΩ(y/|y|)|y|2f(x−y)dy. In this paper, we shall prove that there is a class of functions in H1(S1)−Lln+L(S1)H1(S1)−Lln+L(S1) such that TΩTΩ is weak type L1L1-bounded.