Sobolev Sheaves on Subanalytic Sites

Definitions and mathematical statement

The source asks for a complex of Sobolev sheaves on the subanalytic site of a manifold $M$. In the usual interpretation, $M$ is a real analytic manifold and $M_{\mathrm{sa}}$ has relatively compact subanalytic open subsets as objects, with finite open coverings. The supplied record does not specify the analytic structure, boundary conventions for $M$, or precise site convention; these must be fixed before the question is fully determined.

Write $H^s_{\mathrm{Sob}}(U)$ for the ordinary Sobolev space of regularity $s$ on $U$, separating this notation from the sought sheaf complex $\mathcal S^s$. A relatively compact open subset satisfies $\overline U\subset M$ compact. A Lipschitz open subset is understood to have boundary locally representable as a Lipschitz graph in suitable coordinates. The source does not specify the Sobolev convention, scalar field, auxiliary geometric data, or whether any boundary conditions are intended; none are imposed here.

Existence question. For a manifold $M$ in the intended class and a real regularity parameter $s$, does there exist a complex of sheaves $\mathcal S^s$ on $M_{\mathrm{sa}}$ whose sections on every relatively compact Lipschitz open subset $U\subset M$ are concentrated in degree zero and recover $H^s_{\mathrm{Sob}}(U)$? In particular, the source asks whether such complexes exist throughout the range $s\geq\tfrac12$.

If sections are intended as derived sections, the requested property is \[ R\Gamma(U;\mathcal S^s)\simeq H^s_{\mathrm{Sob}}(U)[0], \] where $[0]$ places the Sobolev space in degree zero. Equivalently, \[ H^j\!\left(R\Gamma(U;\mathcal S^s)\right)\cong \begin{cases} H^s_{\mathrm{Sob}}(U),&j=0,\\ 0,&j\in{\mathbb{Z}}\setminus\{0\}. \end{cases} \] This is a conditional interpretation: the supplied wording does not distinguish derived sections from the complex obtained by taking sections term by term, nor does it specify whether the identifications must respect restriction maps or Sobolev topologies.

Unresolved scope. A relatively compact Lipschitz open subset need not be subanalytic. Consequently, the source's quantifier over every such $U$ requires either a specified extension of evaluation beyond the objects of $M_{\mathrm{sa}}$, or a correction restricting $U$ to subanalytic Lipschitz opens. Neither choice is made here. The record also leaves the class and dimension of $M$, the exact quantification over $s$, and the category and boundedness of the complex unspecified. These qualifications prevent an unconditional precise formulation from being extracted from the record alone.

Short English statement

The question asks whether ordinary Sobolev spaces can be recovered, with no cohomology in other degrees, from a complex of sheaves on a subanalytic site, especially for every $s\geq\tfrac12$. The intended manifold class, Sobolev convention, meaning of sections, and evaluation on Lipschitz opens that are not subanalytic remain unresolved. The supplied record labels the problem partially solved; this draft does not independently verify that assessment.

Sources

• Local registry: data/unsolvedmath/problems.json, with catalog/problems.json as the additional-record registry; supplied permanent UnsolvedMath ID $30003501$, title Sobolev Sheaves on Subanalytic Sites, exact source problem code catalogue entry. Registry-file membership has not been independently checked.
• Supplied report citation: O-Minimality and its Applications to Number Theory and Analysis (2018), Oberwolfach Reports, pp. 1349--1420, DOI 10.4171/owr/2017/22. https://doi.org/10.4171/owr/2017/22. The record attributes the problem to workshop open-problem discussions, supplies proposed year 2017, and names no individual proposer. The differing year indications are retained without reconciliation.
• Supplied literature reference: Lebeau, Sobolev spaces and Sobolev sheaves. https://doi.org/10.24033/ast.1006.
• Supplied literature reference: Oudrane, Sobolev sheaves on the plane. https://arxiv.org/abs/2308.08077.
• Supplied literature reference labelled Oberwolfach Reports 2018. https://doi.org/10.4171/owr/2017/22.
• Literature assessment checked at 2026-08-21: the supplied curation reports Sobolev-sheaf constructions and positive-regularity results in dimension two and related settings, while reporting that existence for all regularities at least one half remains open in general. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. The record's labels partially_solved and corrected_verified, including its assertion that mathematical content was preserved during editorial repairs, are supplied metadata rather than verification by this draft.