On the best constants in Khintchine type inequalities for martingales

For discrete martingale-difference sequences $d=\{d_1,\dots ,d_n\}$ we consider Khintchine type inequalities, involving certain square function $\mathfrak{S}(d)$ considered by Chang-Wilson-Wolff in 1982. In particular, we prove \begin{equation} \left|\sum_{k=1}^nd_k\right|p\le 2^{1/2}\big(Γ((p+1)/2))/\sqrtπ\big)^{1/p}|\mathfrak S(d)|\infty,\quad p\ge 3, \end{equation} where the constant on the right hand side is the best possible and the same as known for the Rademacher sums ${\sum }_{k=1}^n{a}_k{r}_k$. Moreover, for a fixed $n$ the constant in the inequality can be replaced by ${\sum }_{k=1}^n{r}_k\mathrm{/}\sqrt{n}$. We apply a technique, reducing the general case to the case of Haar and Rademacher sums, that allows also establish a sub-Gaussian estimate \begin{equation} {\bf E}\left[\exp\left(λ\cdot \left(\frac{\sum_{k=1}^nd_k}{|\mathfrak S(d)|_\infty}\right)^2\right)\right]\le \frac{1}{\sqrt{1-2λ}},\quad 0<λ<1/2, \end{equation} where the constant on the right hand side is the best possible.