Uniqueness Subsets in Bergman and Korenblum Spaces

Definitions and mathematical statement

The source asks for analogues of previously stated quantitative minorant criteria for separated uniqueness subsets, in two settings: Bergman spaces and growth spaces denoted $A_r^s$ with $0<s<1$. The previous criteria and the definition of $A_r^s$ are not supplied.

In the usual unit-disc setting, let $\mathbb D=\{z\in{\mathbb{C}}:|z|<1\}$, let $\mathcal X$ be the intended space of analytic functions, and let $E\subset\mathbb D$. A uniqueness set for $\mathcal X$ means that $f\in\mathcal X$ and $f|_E=0$ imply $f\equiv0$. Given a radial minorant function $M$, the quantitative property suggested by the cleaned statement is \[ \forall f\in\mathcal X,\qquad \left[\log|f(z)|\leq-M(|z|)\text{ for every }z\in E\right] \ \Longrightarrow\ f\equiv0, \] with $\log0=-\infty$. The required analogues should identify the permitted $E$ and $M$ for which this implication holds, under the source's precise quantifiers and separation assumptions. No necessary-and-sufficient integral condition or numerical threshold is inserted without the missing antecedent results.

For reference, one common separation convention is pseudohyperbolic separation: \[ \inf_{z\ne w\in E}\left|\frac{z-w}{1-\overline w z}\right|>0. \] The intended metric or scale-dependent notion of separation is not specified, nor is whether "uniqueness subset" means that $E$ itself is separated or contains an appropriate separated uniqueness set. Likewise the source does not state monotonicity, regularity or boundary growth conditions on $M$, or whether the inequality permits additive constants, multiplicative bounds or an exceptional finite part of $E$.

Ordinary unweighted Bergman spaces on $\mathbb D$ are $A^p(\mathbb D)=\{f\text{ analytic}:\int_{\mathbb D}|f|^p\,dA<\infty\}$ for $p>0$. This defines a familiar possible setting only; the source does not choose $p$, a weight, normalization or even explicitly reproduce the domain. The growth condition, role and range of $r$ in $A_r^s$ are absent, so that notation is retained rather than identified with an arbitrarily chosen Korenblum class. The restriction $0<s<1$ applies exactly to this second request.

The literature triage checked 2026-08-21 reports these Bergman and fractional Korenblum uniqueness questions unresolved, with partial Korenblum-type results but no full requested characterization. This dated claim and the source-verification note are retained without independent assessment.

Short English statement

Find quantitative conditions under which radial smallness on the intended separated uniqueness subsets forces an analytic function to vanish, for Bergman spaces and for $A_r^s$ with $0<s<1$. The original criterion, separation notion, minorant assumptions and growth-space definitions must be recovered before a sharp analogue can be stated.

Sources

• Local UnsolvedMath registry, permanent ID 30000289, source code catalogue entry, "Uniqueness Subsets in Bergman and Korenblum Spaces," including the original request for analogues, expanded minorant implication and verification metadata.
• Mini-Workshop: Operators on Spaces of Analytic Functions (2006), Oberwolfach Reports, pp. 2043--2090, DOI 10.4171/owr/2005/36: https://doi.org/10.4171/owr/2005/36. The title uses 2006, while the DOI and proposed-year metadata use 2005.
• Supplied reference "recent Korenblum-type results": https://www.i2m.univ-amu.fr/perso/alexandre.boritchev/T36.pdf. This URL is cited as supplied; no PDF has been opened.
• Literature triage checked 2026-08-21; registry record updated 2026-09-16. No external source has been visited or independently checked here.