A weak-type inequality for the martingale square function

Abstract

Let f=(fn)n≥0 be a real-valued martingale satisfying f0≥0 almost surely and let S(f) denote the square function of f. 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 e is the best possible.