Boundary-Supported Interpolating Measures for Bergman Spaces

Definitions and mathematical statement

Let $G\subset{\mathbb{C}}$ be a nonempty bounded planar region (in the usual convention, open and connected). Its Bergman space is \[ L_a^2(G)=\{f\in\mathcal O(G):\int_G|f(z)|^2\,dA(z)<\infty\}, \] with the area $L^2$ norm. Multiplication by the coordinate is the bounded operator $M_zf(z)=zf(z)$ on this space.

The source asks whether there exists such a region for which $M_z$ has a continuous interpolating measure supported on $\partial G$. Under the usual positive finite-measure convention, continuous means atomless: \[ \mu\ne0,\qquad \mu(\{\zeta\})=0\quad\forall\zeta\in{\mathbb{C}}, \qquad \operatorname{supp}\mu\subseteq\partial G. \] Support contained in the boundary does not require equality with the whole boundary. The exact operator-theoretic definition of "interpolating measure" is not supplied, and neither are positivity, finiteness or normalization conventions; these must be checked rather than silently fixed by the displays.

Thus the requested existence concerns a boundary-supported, non-atomic measure satisfying the source's interpolation property for $M_z$. It cannot be replaced by existence of any boundary measure, a Carleson measure, an interpolating sequence, or a normal-extension representing measure without a theorem identifying those notions. If the interpolation definition uses boundary values of functions, the relevant trace/functional-calculus interpretation is also necessary because no boundary regularity of $G$ is assumed. These omissions prevent a complete independent predicate-level formulation of interpolation from this excerpt alone.

The supplied literature assessment checked 21 August 2026 reports this boundary-supported continuous interpolation problem still open and listed without resolution in later literature. That dated curation is retained as a source qualification, not an independently verified current-status claim.

Short English statement

Does some bounded planar region have a Bergman multiplication operator admitting an atomless interpolating measure concentrated on its boundary? The specialized interpolation definition and measure conventions are missing. The supplied assessment reports that later problem lists still leave the question open.

Sources

• Local UnsolvedMath registry, permanent ID 30000288, exact source code catalogue entry; Boundary-Supported Interpolating Measures for Bergman Spaces. Proposed year 2005, status open; corrected_verified statement preserves the bounded region, Bergman operator, continuity and boundary support.
• Supplied report citation: Mini-Workshop: Operators on Spaces of Analytic Functions (2006), Oberwolfach Reports, DOI 10.4171/owr/2005/36, pp. 2043--2090, https://doi.org/10.4171/owr/2005/36. Title year differs from DOI/proposed year.
• Supplied literature triage checked 2026-08-21: "original OWR problem", https://doi.org/10.4171/owr/2005/36; "later OWR problem list", https://ems.press/content/serial-article-files/46010. This dated OWR-derived curation requests independent verification of time-sensitive status and cited claims.