Polynomial Density on Crescent Domains

Definitions and mathematical statement

Let $\Omega\subset{\mathbb{C}}$ be a domain described in the supplied source as a "crescent domain" or "crescent-type domain," with bounded complementary component $G$ and unbounded complementary component $\Omega_\infty$. The source record does not define the crescent condition or specify the boundary regularity it entails; that geometric qualification must therefore be retained pending clarification. Set \[ \delta(z)=\operatorname{dist}(z,\Omega_\infty) =\inf_{w\in\Omega_\infty}|z-w|. \] Let $\omega$ be the source's harmonic measure on $\partial G$. Its defining domain, side of the boundary, and pole are not specified in the supplied record.

Write $dA$ for planar area measure and, for $1\leq p<\infty$, define \[ L_a^p(\Omega,dA)=\left\{f:\Omega\to{\mathbb{C}}:\ f\text{ is holomorphic and }\int_\Omega |f(z)|^p\,dA(z)<\infty\right\}. \] The source asks whether \[ \int_{\partial G}\log\delta(z)\,d\omega(z)=-\infty \quad\Longrightarrow\quad H^p(\Omega,dA)=L_a^p(\Omega,dA) \quad\text{for every }1\leq p<\infty. \] Here $\log 0=-\infty$, and the integral condition is understood as an extended Lebesgue integral equal to $-\infty$, rather than an undefined difference of infinite quantities.

The supplied record does not define $H^p(\Omega,dA)$. Its title and literature assessment indicate that polynomial density is intended. To express that interpretation explicitly, put \[ \mathcal P_p(\Omega)=\{P|_\Omega:P\in{\mathbb{C}}[z],\ P|_\Omega\in L^p(\Omega,dA)\}. \] If the intended definition is \[ H^p(\Omega,dA)=\overline{\mathcal P_p(\Omega)}^{\,L^p(\Omega,dA)}, \] then the proposed conclusion says that, for every real $p$ with $1\leq p<\infty$, every $f\in L_a^p(\Omega,dA)$, and every $\varepsilon>0$, there exists a polynomial $P$ whose restriction belongs to $L^p(\Omega,dA)$ such that \[ \left(\int_\Omega |f(z)-P(z)|^p\,dA(z)\right)^{1/p}<\varepsilon. \] This interpretation of $H^p$ is conditional and requires confirmation from the source, together with the precise crescent-domain and harmonic-measure conventions.

Short English statement

The question asks whether divergence to $-\infty$ of the harmonic-measure integral of the logarithmic distance to the outer complementary component forces polynomial density in every finite-exponent Bergman space. The crescent geometry, harmonic measure, and notation $H^p$ require clarification before this interpretation is fully specified.

Sources

• Local UnsolvedMath registry: data/unsolvedmath/problems.json, permanent ID $30001612$, source problem code catalogue entry, title "Polynomial Density on Crescent Domains," in the Oberwolfach Reports Open Problems collection. No individual proposer is identified in the supplied record.
• Supplied report citation: Operator Theory and Harmonic Analysis (2011), Oberwolfach Reports, pp. 2813--2875, DOI 10.4171/owr/2010/49. https://doi.org/10.4171/owr/2010/49. The record assigns proposed year 2010. Its corrected_verified label and assertion that the statement was checked against the official report are supplied verification metadata, not an independent verification here.
• Supplied literature reference: "Hedenmalm and Shimorin weighted Bergman estimates," https://arxiv.org/abs/math/0406345. No fuller bibliographic details or theorem locator are supplied.
• Supplied literature reference: "original problem report," https://doi.org/10.4171/owr/2010/49.
• Literature checked at: 2026-08-21. This is supplied dated curation, not independently verified. The supplied status is "partially solved." The accompanying assessment states that the divergence condition implies polynomial density for $1\leq p<3+\tau$, for a universal constant $\tau>0$, while the assertion for every finite $p$ remains unresolved. The background describes this as dated literature triage derived from the supplied Oberwolfach curation and requests independent verification of time-sensitive status and cited claims.