Characterizations of the UMD property via tail estimates for tangent processes
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2026
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| _version_ | 1866918492845899776 |
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| author | Bodó, Gergely Yaroslavtsev, Ivan |
| author_facet | Bodó, Gergely Yaroslavtsev, Ivan |
| contents | We characterize the UMD property of a Banach space by tail inequalities for maximal functions of tangent conditionally symmetric processes. More precisely, we prove that a Banach space $V$ is UMD if and only if for some (equivalently, for all) $p\in(0,\infty)$ one has that \[ \mathbb P(\sup_{r\geq 0} \| N_r\|>t)\lesssim_{p,V}\Bigl(\frac{s^p}{t^p}+\mathbb P(\sup_{r\geq 0} \| M_r\|>s)\Bigr), \qquad s,t>0, \] for all tangent conditionally symmetric $V$-valued processes $M$ and $N$. We further show that this estimate is equivalent to suitable Lorentz norm inequalities for the associated maximal functions, and obtain analogous characterizations in the discrete-time, continuous-time, and purely discontinuous settings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_09177 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Characterizations of the UMD property via tail estimates for tangent processes Bodó, Gergely Yaroslavtsev, Ivan Functional Analysis Probability Primary 60B11, Secondary 46G10, 60H05 We characterize the UMD property of a Banach space by tail inequalities for maximal functions of tangent conditionally symmetric processes. More precisely, we prove that a Banach space $V$ is UMD if and only if for some (equivalently, for all) $p\in(0,\infty)$ one has that \[ \mathbb P(\sup_{r\geq 0} \| N_r\|>t)\lesssim_{p,V}\Bigl(\frac{s^p}{t^p}+\mathbb P(\sup_{r\geq 0} \| M_r\|>s)\Bigr), \qquad s,t>0, \] for all tangent conditionally symmetric $V$-valued processes $M$ and $N$. We further show that this estimate is equivalent to suitable Lorentz norm inequalities for the associated maximal functions, and obtain analogous characterizations in the discrete-time, continuous-time, and purely discontinuous settings. |
| title | Characterizations of the UMD property via tail estimates for tangent processes |
| topic | Functional Analysis Probability Primary 60B11, Secondary 46G10, 60H05 |
| url | https://arxiv.org/abs/2605.09177 |