The Gundy-Stein decomposition with explicit constants

Abstract: Let $(\mathcal F_n){n\ge 1}$ be a filtration and let $f\ge0$ belong to $L^1(\mathcal F\infty)$. For the martingale $f_n=\mathbb E[f\mid \mathcal F_n]$ and each $λ>0$ 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 $λ$. 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 $0\le k\le λ$. 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 $α$-regular filtrations.