In joint work with C. Thiele, we prove that the bilinear Hilbert transform maps into weak L^{2/3}, up to a doubly logarithmic factor. The main technical advancement we employ in the proof is a strengthening of the multi-frequency Calderon-Zygmund decomposition of Nazarov, Oberlin and Thiele where, loosely speaking, the interaction of the bad part, i.e. having mean-zero with respect to N frequencies, with functions localized near one of these frequencies is exponentially small in terms of the good, i.e. L^2, part.
Via the same techniques, we also investigate the sharp behavior of weak type L^p bounds near p=1 for the Carleson operator and its lacunary version, which is intimately and directly connected to Konyagin’s conjectures on pointwise convergence of Fourier series in endpoint Lorentz-Orlicz spaces near L^1.
A further application we explore, in joint work with Andrei Lerner, is to sharp weighted bounds for Carleson-type operators in terms of the A_p constant of the weight.