PP* estimates for the truncated Carleson operator

We consider two Hilbert transforms, with real phases a and b, smoothly truncated at ${{2^{-k_0}}}$ where k ₀ is a non-negative integer. We decode the operator resulting from composing one of them with the adjoint of the other one. Then the case of a similarly truncated Carleson operator is dealt with. The case of Hilbert transforms, as well as the Carleson operator, truncated away from the origin is also considered. An application is mentioned.