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PreprintarXiv licencearXiv

๐’œ-free measures and the Rank-one Theorem in Carnot Groups

  • Guido De Philippis
  • Alberto Gervani
  • Annalisa Massaccesi
  • Davide Vittone
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Abstract

We establish a structure theorem for measures satisfying left-invariant PDE constraints in Carnot groups. Our generalization, that takes inspiration from the paper arxiv:1601.06543 by the first named author and F. Rindler, requires the introduction of the hypoelliptic wave cone, a sub-Riemannian version of the wave cone from the theory of Compensated Compactness, and it employs classical results in Harmonic Analysis on homogeneous spaces. Then, we utilize this result to extend the Rank-one Theorem for functions of bounded horizontal variation (BVH) to all Carnot groups. This is achieved by considering a suitable curl-type left-invariant operator, that provides necessary differential constraints for horizontal gradients, and by studying the hypoellipticity of said constraints.

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Cite

BibTeXRIS
BibTeX
@misc{philippis2026๐’œfree,
  title = {{$\mathscrA$-free measures and the Rank-one Theorem in Carnot Groups}},
  author = {Guido De Philippis and Alberto Gervani and Annalisa Massaccesi and Davide Vittone},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.25895},
  eprint = {2609.25895},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.25895},
}
RIS
TY  - UNPB
TI  - ๐’œ-free measures and the Rank-one Theorem in Carnot Groups
AU  - Guido De Philippis
AU  - Alberto Gervani
AU  - Annalisa Massaccesi
AU  - Davide Vittone
PY  - 2026
DA  - 2026/09/22
PB  - arXiv
DO  - 10.48550/arxiv.2609.25895
UR  - https://arxiv.org/abs/2609.25895
AB  - We establish a structure theorem for measures satisfying left-invariant PDE constraints in Carnot groups. Our generalization, that takes inspiration from the paper arxiv:1601.06543 by the first named author and F. Rindler, requires the introduction of the hypoelliptic wave cone, a sub-Riemannian version of the wave cone from the theory of Compensated Compactness, and it employs classical results in Harmonic Analysis on homogeneous spaces. Then, we utilize this result to extend the Rank-one Theorem for functions of bounded horizontal variation (BV_H) to all Carnot groups. This is achieved by considering a suitable curl-type left-invariant operator, that provides necessary differential constraints for horizontal gradients, and by studying the hypoellipticity of said constraints.
ER  -