The Gundy-Stein decomposition with explicit constants

TL;DR

This paper presents an explicit Gundy-Stein decomposition for martingales with detailed bounds, sharpness results, and applications to weak-type estimates and BMO inequalities.

Contribution

It introduces a new Gundy-Stein decomposition with explicit constants, sharpness results, and refined four-term decomposition, advancing martingale analysis techniques.

Findings

Explicit bounds for the decomposition parts.

Sharpness results for the decomposition under certain conditions.

Applications to weak-type (1,1) estimates and BMO inequalities.

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 $\lambda>0$ 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.