A weak-type inequality for the martingale square function

Let f=(fn)n≥0f=(fn)n≥0 be a real-valued martingale satisfying f0≥0f0≥0 almost surely and let S(f)S(f) denote the square function of ff. The paper contains the proof of the weak-type bound λP(S(f)≥λ)≤eEsupn≥0fn,λ>0, involving the one-sided maximal function on the right-hand side. The constant ee is the best possible.