The Gundy-Stein decomposition with explicit constants

Abstract

Let $({\mathcal{F}}_n{)}_{n\ge 1}$ be a filtration and let $f\ge 0$ belong to $L^1({\mathcal{F}}_{\infty })$. For the martingale $f_n=\mathbb{E}[f\mid {\mathcal{F}}_n]$ and each $\lambda >0$ we prove a Gundy--Stein decomposition \[ f=g+h+k \] with explicit numerical constants. In the positive closed case the three parts satisfy explicit bounds, and the bounded part is bounded above by $\lambda$. We also prove a one-parameter form for the bounded part and two-point sharpness results, including a joint sharpness statement for arbitrary decompositions under the condition $0\le k\le \lambda$. We also obtain an exact four-term refinement of the decomposition, separating the bounded term into a stopped part and a conditional expectation term. As applications we obtain an explicit weak-type $(1,1)$ estimate for truncated martingale multipliers and a John--Nirenberg inequality for martingale $\mathrm{BMO}$ on atomic $\alpha$-regular filtrations.