A weak-type inequality for the martingale square function (Q464485)

Let \(f=(f_n)_{n\geq 0}\) be a real valued martingale with \(f_0 \geq 0\) almost surely. Define the associated difference sequence \(df_0 = f_0\), \(df_n = f_n - f_{n-1}\), \(n=1,2, \dots\) and the square function \(S(f) = \left( \sum_{n=0}^{\infty} |df_n|^2 \right) ^{1/2}\). This note establishes a bound involving the one-sided maximal function \(f^* = \sup_{n \geq 0} \;f_n\) which is a variation on the bound established in [\textit{D. C. Cox}, Proc. Am. Math. Soc. 85, 427--433 (1982; Zbl 0494.60047)]. The result shows that for any \(\lambda >0\),\ \(\lambda \operatorname{P}(S(f) \geq \lambda) \leq e \operatorname{E}( f^*)\), and that the inequality is sharp.