Sufficiency of the operators \(\mathscr A^s\) for characterizing horizontal gradients

Establish whether, after removing suitable redundancies, the curl-type operators \(\mathscr A^s\) constructed for Carnot groups provide sufficient conditions for a measure to be a horizontal gradient, either in general Carnot groups or at least in free Carnot groups.

Background

The paper constructs, for a Carnot group of step ss, a left-invariant differential operator As\mathscr A^s such that every horizontal gradient DHuD_Hu is As\mathscr A^s-free. This necessary differential constraint is then used to prove the rank-one theorem for singular parts of horizontal derivatives.

The unresolved issue is whether these operators characterize horizontal gradients, rather than merely providing necessary conditions, once proper redundancies among their components are removed. The authors note that this sufficiency holds in the first Heisenberg group H1H^1 for suitable components of A2\mathscr A^2, but state that they did not investigate the corresponding question for general Carnot groups or free Carnot groups.

References

One might wonder whether, after getting rid of some proper redundancies, our operators $\mathscr As$ provide also sufficient conditions in general Carnot groups, or at least in free ones (see §9.1.1 and §11.1.3.6). We did not investigate this question as it was not among the scopes of the present paper; let us mention, however, that this is true in $H1 $ by considering certain components of $\mathscr A2$ -- namely, the operator\footnote{This operator, by the way, coincides with the well-known one introduced by M.~Rumin.} $(\mathscr{A}2_{(1,1,2)},\mathscr{A}2_{(2,1,2)})$, see Remark~\ref{rem_rumin}.

— $\mathscr{A}$-free measures and the Rank-one Theorem in Carnot Groups  (2609.25895 - Philippis et al., 22 Sep 2026) in Remark immediately following Theorem 1.2 in Section 1 (Introduction)