Heat content asymptotics for irregular domains in general Carnot groups

Determine whether the heat content asymptotics for the fractional sub-Laplacian on bounded open sets in general Carnot groups hold without imposing regularity conditions on the domain boundary, including the case of the ordinary sub-Laplacian with fractional parameter α=2.

Background

The paper establishes small-time relative heat content asymptotics for fractional sub-Laplacians under boundary regularity assumptions, and for 0<α<1 it also treats bounded open sets with finite fractional horizontal perimeter. The authors note that, in step-two Carnot groups, related asymptotics are known for bounded finite-perimeter sets without boundary regularity assumptions. However, the argument used in the step-two case does not extend to arbitrary Carnot groups, leaving unresolved whether analogous heat content asymptotics remain valid for bounded open sets in general Carnot groups when no regularity is assumed for the boundary. The question includes the classical sub-Laplacian case α=2.

References

It still remains an open question whether the heat content asymptotics hold (even when $\alpha=2$) for bounded open sets of general Carnot groups without assuming any regularity conditions on the boundary.

— Heat content and spectrum for subordinated sub-Laplacians  (2609.04045 - Gordina et al., 3 Sep 2026) in Remark following Theorem 2.4, Section 2.3 (Relative heat content of subordinated sub-Laplacian)