Dimension-Independent Analyticity Bounds for K-Convex Spaces

Definitions and mathematical statement

A Banach space $X$ is $K$-convex if its Rademacher projections are uniformly bounded on $L^2$: explicitly, let $\Omega_n=\{-1,1\}^n$ with uniform probability measure, let $\varepsilon_j$ be the coordinate functions, and set \[ P_n f=\sum_{j=1}^n\varepsilon_j\,{\mathbb{E}}_{\Omega_n}[\varepsilon_j f], \qquad f:\Omega_n\to X. \] Then $K$-convexity means that there is a finite constant $K$ such that \[ \|P_n f\|_{L^2(\Omega_n;X)}\le K\|f\|_{L^2(\Omega_n;X)} \quad\text{for every }n\ge1\text{ and every }f:\Omega_n\to X. \] The source introduces the sector \[ \Sigma_\theta=\{z\in{\mathbb{C}}\setminus\{0\}:|\arg z|<\theta\}. \] However, the supplied record does not define $M_{p,d,\theta}^X$, the operators or functions to which these bounds apply, or the admissible range of $\theta$. Consequently, the following is a quantified transcription of the supplied question, conditional on recovering that missing definition.

For every $K$-convex Banach space $X$ and every real $p$ with $1<p<\infty$, do there exist an admissible angle $\theta=\theta(X,p)>0$ and a finite constant $C=C(X,p,\theta)$ such that \[ M_{p,d,\theta}^X\le C \qquad\text{for every }d\in\{1,2,3,\ldots\}, \] or equivalently, \[ \sup_{d\ge1}M_{p,d,\theta}^X<\infty? \] Here the angle and bound may depend on $X$ and $p$, but must be independent of $d$, as in the supplied cleaned statement.

Unresolved source qualifications. The original excerpt also contains an angle restriction $0<\theta\le\theta_p$, without defining $\theta_p$. It attributes a $K$-convexity characterization to Pisier, reference [10], but its exponent quantifier is corrupted as "for some all (equivalently: for some)." This prevents a reliable comparison between that characterization and the proposed question. The excerpt further reports that adapting an argument of reference [4] gives bounds growing exponentially in $d$; neither that reference nor the underlying estimate is specified in the supplied record. These fragments do not supply the missing definition of $M_{p,d,\theta}^X$.

Short English statement

The supplied question asks whether every $K$-convex Banach space admits, for each $1<p<\infty$, an angle and a bound that work uniformly in every positive integer dimension. The quantity being bounded is undefined in the record, and a related characterization has corrupted quantifiers, so the intended analytic statement requires source clarification. The supplied status is open; this is not an independently verified assessment.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30006501$, title: Dimension-Independent Analyticity Bounds for K-Convex Spaces; source problem code catalogue entry, identified in the excerpt as Problem 2. No proposer is named; the supplied proposed year is 2025.
• Supplied source citation: Recent Developments in SPDEs and BSDEs meet Harmonic and Functional Analysis (2026), Oberwolfach Reports, DOI 10.4171/owr/2025/53, pp. 2843--2916. The excerpt includes the page marker 2897. https://doi.org/10.4171/owr/2025/53. The background describes Oberwolfach Reports as documenting workshops and their open-problem discussions.
• Supplied literature source: Oberwolfach Report 53/2025. https://doi.org/10.4171/owr/2025/53.
• 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 open and reports that dimension-dependent estimates are available while the proposed dimension-independent bounds for all $K$-convex spaces and all $p$ remain an explicit open problem. This is supplied dated curation derived from the OWR record, not independently verified literature or source verification; its status and cited claims require independent checking.
• The original excerpt attributes a characterization to Pisier [10] and an argument to [4]. Their bibliographic entries are absent, so these reference labels cannot be resolved from the supplied evidence.