Several Complex Variables Problem List — Problem 21

Definitions and mathematical statement

For a positive integer $n$, let \[ B_n=\{z\in{\mathbb{C}}^n:\|z\|_2<1\},\qquad S_n=\partial B_n. \] The supplied question is: characterize the representing measures for evaluation at $0$ in the unit ball. The source does not specify the class of test functions, where the measures are supported, or the convention on positivity, so it does not uniquely determine the family to be characterized.

One possible precise interpretation uses the ball algebra \[ A(B_n)=\{f\in C(\overline{B_n};{\mathbb{C}}):f|_{B_n}\text{ is holomorphic}\}. \] For a specified choice $X\in\{B_n,\overline{B_n},S_n\}$, define \[ \mathcal M_X(0;A(B_n))= \left\{\mu:\begin{array}{l} \mu\text{ is a positive Borel probability measure on }X,\\[2pt] \displaystyle\int_X f|_X\,{\,\mathrm{d}}\mu=f(0)\quad\text{for every }f\in A(B_n) \end{array}\right\}. \] Under this interpretation, the problem is to give a necessary and sufficient characterization of membership in $\mathcal M_X(0;A(B_n))$, for the intended support set $X$. This is a conditional formulation: the ball algebra, positivity convention, and choice of $X$ are not specified in the supplied record and must be confirmed before this can be treated as the intended mathematical statement.

Short English statement

Describe exactly which measures reproduce evaluation at the origin by integration against the intended test functions on the unit ball in ${\mathbb{C}}^n$. The supplied record leaves the test functions, support of the measures, and positivity convention unspecified; the displayed ball-algebra formulation is only a possible interpretation.

Sources

• Local UnsolvedMath registry, permanent ID $800225$, source problem code catalogue entry, title: "Several Complex Variables Problem List --- Problem 21." The supplied registry labels the problem open; this status is not independently verified here.
• Buzzard, Fornaess, Krantz, The Several Complex Variables Problem List (1995), Problem 21. Supplied source URL: https://arxiv.org/abs/math/9509202. The registry attributes the problem to these authors and gives the proposed year as 1995.
• Supplied provenance records access on 2026-10-16 and extraction from arXiv LaTeX. These are supplied provenance details, not a claim of source access or verification in preparing this draft.