Uniqueness of Dirichlet Forms on the Sierpinski Carpet

Definitions and mathematical statement

Let $F$ be the Sierpiński carpet intended by the source, and let $\mu$ be the common reference measure for its two constructions. The classical planar carpet is obtained by repeatedly removing the open middle square from each retained square of a $3\times3$ subdivision; the extract does not specify this example versus a generalized carpet class. A symmetric Dirichlet form on $L^2(F,\mu)$ is a densely defined nonnegative symmetric bilinear form $(\mathcal E,\mathcal D)$, closed for the norm $(\|u\|_2^2+\mathcal E(u,u))^{1/2}$, with the Markov contraction property. Write $(\mathcal E_{\mathrm{BB}},\mathcal D)$ and $(\mathcal E_{\mathrm{KZ}},\mathcal D)$ for the Barlow--Bass and Kusuoka--Zhou forms, whose common domain the source calls Besov--Lipschitz. The precise measure and Besov parameters are not supplied. The first question is \[\exists c>0\ \forall u,v\in\mathcal D:\qquad \mathcal E_{\mathrm{BB}}(u,v)=c\,\mathcal E_{\mathrm{KZ}}(u,v).\] This is equality up to one global positive scalar, not merely equality of domains or comparability of the two energies. The source describes it equivalently as uniqueness, up to positive rescaling, of a nice Dirichlet form on $F$. The admissible nice class is not defined: regularity, locality, symmetry/self-similarity assumptions, and boundary behavior must be specified before that uniqueness assertion is precise. Equality of these two named forms alone is not logically equivalent to uniqueness among an undefined larger class, so the claimed equivalence remains a source qualification requiring its surrounding hypotheses. The 2026-08-21 triage labels the record open and says the cited constructions do not settle the uniqueness question; this supplied assessment is preserved without a new literature conclusion.

Short English statement

Do the two named carpet energies agree up to one positive constant on their common domain? The accompanying uniqueness question needs a precise definition of nice, a fixed reference measure, and the intended carpet and boundary conventions.

Sources

• Local UnsolvedMath registry, permanent ID 30000967, source problem catalogue entry, Uniqueness of Dirichlet Forms on the Sierpinski Carpet, proposed year 2008; no individual proposer supplied. The statement names the Barlow--Bass and Kusuoka--Zhou constructions.
• Supplied citation: Stochastic Analysis (2009), Oberwolfach Reports, DOI 10.4171/owr/2008/25, pp. 1361--1424: https://doi.org/10.4171/owr/2008/25. Supplied literature label: original Oberwolfach report.