The Gundy-Stein decomposition with explicit constants

Let $({\mathcal{F}}_n{)}_{n\ge 1}$ be a filtration and let $f\ge 0$ belong to $L^1({\mathcal{F}}_{\mathrm{\infty }})$. 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 $\mathrm{B}\mathrm{M}\mathrm{O}$ on atomic $\alpha$-regular filtrations.