Exponential Boundary Growth in the Dirichlet Weak Product

Definitions and mathematical statement

Let $\mathbb D=\{z\in{\mathbb{C}}:|z|<1\}$. The Dirichlet space without constants, with the normalization specified by the supplied orthonormal basis, is \[ D_0=\left\{f(z)=\sum_{n=1}^{\infty}a_nz^n\in\operatorname{Hol}(\mathbb D):\ \|f\|_{D_0}^2=\sum_{n=1}^{\infty}n|a_n|^2<\infty\right\}. \] Thus $(n^{-1/2}z^n)_{n\geq1}$ is an orthonormal basis of $D_0$. Following the supplied record, let $D_0^{\odot2}$ consist of functions admitting a representation \[ F(z)=\sum_{j=1}^{\infty}t_jf_j(z)g_j(z),\qquad t_j\geq0,\qquad \sup_{j\geq1}t_j<\infty, \] where $(f_j)_{j\geq1}$ and $(g_j)_{j\geq1}$ are each orthonormal bases of $D_0$. The coefficients are required to be bounded; the supplied definition does not require their summability.

The source asks for the boundary behavior of functions in $D_0^{\odot2}$, and particularly for the fastest possible growth, as $r\to1^-$ with $0<r<1$, of the expression \[ \int_{|z|=r}\exp\!\left(\frac{|F(z)|^2}{\log(1/(1-r^2))}\right)\,dA, \qquad F\in D_0^{\odot2},\quad \|F\|<1. \] Here $dA$ is retained from the original statement. The supplied record does not define a norm on $D_0^{\odot2}$, so the condition $\|F\|<1$ remains unresolved. It also does not define the measure in this integral: the original statement writes $dA$ on a circle, whereas the clean statement omits the differential. Planar area measure, arclength measure, and normalized angular measure cannot be silently identified. Finally, the record does not specify whether the requested growth concerns each fixed admissible $F$, a supremum over admissible $F$ at each radius, or another sharp asymptotic formulation. These qualifications prevent a uniquely specified extremal problem without further source clarification.

Short English statement

The supplied problem asks how functions represented by bounded nonnegative coefficients and products of two orthonormal bases of the zero-constant Dirichlet space behave near the unit circle. Its specific target is sharp growth of an exponential boundary integral under $\|F\|<1$. The norm, integration measure, and precise asymptotic quantifiers require clarification. The supplied open status is dated curation, not an independently verified conclusion.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001615$, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The supplied title is "Exponential Boundary Growth in the Dirichlet Weak Product." No individual proposer is identified.
• Supplied source citation: Operator Theory and Harmonic Analysis (2011), Oberwolfach Reports, DOI 10.4171/owr/2010/49, pp. 2813--2875. https://doi.org/10.4171/owr/2010/49. The record assigns proposed year 2010 and describes the report as documenting workshop open-problem discussions; these bibliographic details are retained as supplied.
• Supplied literature source: "Arcozzi Rochberg Sawyer and Wick Dirichlet survey." https://arxiv.org/abs/1008.5342.
• Supplied literature source: "original problem report." https://doi.org/10.4171/owr/2010/49.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that Dirichlet-space surveys and related estimates provide context, but that no result resolving the requested exact asymptotic was found. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment; time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports that it was checked against the official Oberwolfach report. This is a supplied verification claim; no independent source consultation or mathematical verification is asserted here.