Weak type inequalities for conditionally symmetric martingales

Let ff be a conditionally symmetric martingale and let S(f)S(f) be its square function. We prove that ‖f‖p,∞≤Cp‖S(f)‖p,1≤p≤2, where Cpp=21−p/2πp−3/2Γ((p+1)/2)Γ(p+1)1+132+152+172+⋯1−13p+1+15p+1−17p+1+⋯. In addition, the constant CpCp is shown to be the best possible even for the class of dyadic martingales.