New estimates for the maximal singular integral

In this paper we pursue the study of the problem of controlling the maximal singular integral T ∗f by the singular integral Tf . Here T is a smooth homogeneous Calderón-Zygmund singular integral of convolution type. We consider two forms of control, namely, in the L2(Rn) norm and via pointwise estimates of T ∗f by M(Tf) or M2(Tf) , where M is the Hardy-Littlewood maximal operator and M2 = M ◦M its iteration. It is known that the parity of the kernel plays an essential role in this question. In a previous article we considered the case of even kernels and here we deal with the odd case. Along the way, the question of estimating composition operators of the type T ⋆◦T arises. It turns out that, again, there is a remarkable difference between even and odd kernels. For even kernels we obtain, quite unexpectedly, weak (1, 1) estimates, which are no longer true for odd kernels. For odd kernels we obtain sharp weaker inequalities involving a weak L1 estimate for functions in LLogL.