The best constant in Burkholder's weak-$L\sp{1}$ inequality for the martingale square function

Let ${Y_1},{Y_2}, \ldots$ be a martingale with difference sequence ${X_1} = {Y_1},{X_i} = {Y_i} - {Y_{i - 1}},i \geqslant 2$. We give a new proof of the inequality

$$P\left( {\sum\limits_{i \geqslant 1} {X_i^2 \geqslant {\lambda ^2... ...{ - 1}}C\mathop {\sup }\limits_{i \geqslant 1} E\left\vert {{Y_i}} \right\vert,$$

for all $y > 0$, and show that the best constant is $C = {e^{1/2}}$.