The Gundy-Stein decomposition with explicit constants
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866908920751063040 |
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| author | Hormozi, Mahdi Zhu, Jie-Xiang |
| author_facet | Hormozi, Mahdi Zhu, Jie-Xiang |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_28226 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Gundy-Stein decomposition with explicit constants Hormozi, Mahdi Zhu, Jie-Xiang Probability 60G44, 60G46, 46E30 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. |
| title | The Gundy-Stein decomposition with explicit constants |
| topic | Probability 60G44, 60G46, 46E30 |
| url | https://arxiv.org/abs/2603.28226 |