Dimension-Free Bounds and Optimal Ornstein-Uhlenbeck Calculus Angles

Definitions and mathematical statement

Let $X$ be a Banach space. In standard terminology, $X$ is $K$-convex if its finite Rademacher projections are uniformly bounded: on $\Omega_n=\{-1,1\}^n$ with uniform probability measure, put $r_j(\omega)=\omega_j$ and \[ P_n f=\sum_{j=1}^n r_j\,{\mathbb{E}}(r_j f). \] Then $K$-convexity means \[ \sup_{n\geq1}\|P_n\|_{L^2(\Omega_n;X)\to L^2(\Omega_n;X)}<\infty. \] The space $X$ has the UMD property if, for some $q\in(1,\infty)$, there is $\beta<\infty$ such that every finite $X$-valued martingale difference sequence $(d_j)_{j=1}^n$ in $L^q$, on any probability space with a filtration, and every choice $\varepsilon_j\in\{-1,1\}$, satisfy \[ \left\|\sum_{j=1}^n\varepsilon_j d_j\right\|_{L^q(X)} \leq \beta\left\|\sum_{j=1}^n d_j\right\|_{L^q(X)}. \] For an open set $\Omega\subseteq{\mathbb{C}}$, $H^\infty(\Omega)$ denotes the bounded holomorphic functions on $\Omega$, equipped with the supremum norm.

Unresolved source notation. The supplied record does not define $M_{p,d,\theta}^X$, the angle $\theta_p$, or the region $\Sigma_\theta$. It identifies $-L_X$ as an Ornstein--Uhlenbeck operator but does not specify its ambient function space, underlying measure, dimension, normalization, or operator domain. The scalar field of $X$ and any complexification needed for holomorphic functional calculus are also unspecified. These omissions prevent a fully determined mathematical formulation. The following questions therefore retain the source symbols as unresolved notation.

Dimension-free bounds. For every $K$-convex Banach space $X$, is it true that \[ \forall p\in(1,\infty)\quad \forall\theta\in(0,\theta_p)\quad \sup_{d\in\{1,2,\ldots\}}M_{p,d,\theta}^X<\infty? \] The additional endpoint question is whether, for every such $X$, \[ \forall p\in(1,\infty)\quad \sup_{d\in\{1,2,\ldots\}}M_{p,d,\theta_p}^X<\infty. \] The requested bounds are uniform in $d$; no uniformity in $X$, $p$, or $\theta$ is asserted.

Functional calculus. For every UMD Banach space $X$, is it true that \[ \forall p\in(1,\infty)\quad \forall\theta\in(0,\theta_p)\quad -L_X\text{ has a bounded }H^\infty(\Sigma_\theta)\text{ functional calculus}? \] The dependence of the realization of $-L_X$ on $p$ must be specified before this question is fully defined. The inequalities $0<\theta<\theta_p$ are retained exactly as supplied; the sector convention and the meaning of the claimed optimal angle require clarification. No endpoint functional-calculus assertion is explicitly requested. In all questions, $p=1$, $p=\infty$, and $\theta=0$ are excluded.

Short English statement

The source asks whether $K$-convexity gives bounds uniform over all positive integer dimensions, for every $1<p<\infty$ and $0<\theta<\theta_p$, and possibly also at $\theta=\theta_p$. It separately asks whether UMD spaces give the stated Ornstein--Uhlenbeck $H^\infty$ calculus throughout the same strict angle range. Essential notation and operator domains are missing, so these are conditional transcriptions requiring mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30006502$, title Dimension-Free Bounds and Optimal Ornstein-Uhlenbeck Calculus Angles, exact source problem code catalogue entry. The record attributes the problem to Oberwolfach Reports workshop open-problem discussions; no individual proposer is supplied. The proposed year is recorded as 2025.
• Supplied source citation: Recent Developments in SPDEs and BSDEs meet Harmonic and Functional Analysis (2026), Oberwolfach Reports, pp. 2843--2916, DOI 10.4171/owr/2025/53. The literature-source label is Oberwolfach Report 53/2025. https://doi.org/10.4171/owr/2025/53. The raw extracted statement contains a page marker 2897; this has not been independently checked as the precise problem location.
• Supplied literature source: Holomorphic functional calculus and vector-valued Littlewood--Paley theory. https://doi.org/10.4171/JEMS/1430.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem partially_solved and reports that a bounded $H^\infty$ calculus is known for UMD spaces at some suboptimal sector angle, while the requested optimal-angle statement remains open. This is supplied dated curation, not an independently verified assessment. The background expressly requests independent verification of time-sensitive status and cited claims.