On the best constants in Khintchine type inequalities for martingales

For discrete martingale-difference sequences d = { d 1 , … , d n } d=\{d_1,… ,d_n\} we consider Khintchine type inequalities, involving certain square function S ( d ) S(d) introduced by Chang-Wilson-Wolff [Comment. Math. Helv. 60 (1985), pp. 217–246]. In particular, we prove ‖ ∑ k = 1 n d k ‖ p ≤ 2 1 / 2 ( Γ ( ( p + 1 ) / 2 ) ) / π ) 1 / p ‖ S ( d ) ‖ ∞ , p ≥ 3 , {equation*} \|∑ ₖ₌₁^nd_k \|_p≤ 21/2 (Γ ((p+1)/2))/√π )1/p\|{ S}(d)\|_∞ , p≥ 3, {equation*} where the constant on the right hand side is optimal and matches the one known for Rademacher sums ∑ k