Obstructions to exact regularity of sublaplacians
Abstract
Let M be a closed manifold equipped with a collection of smooth real vector fields X1,…, Xk and a smooth measure μ. Let $P:=\sum_j X_j^\dagger X_j $ be the associated sublaplacian. Under the assumption that each pair of points of M can be connected by a path obtained by joining integral curves of the vector fields, the equation Pu=f admits unique zero-average solutions u in the natural energy space, for all zero average data f∈ L2. We say that exact regularity holds for the above equation if u∈ Hk(M) whenever f∈ Hk(M), where Hk(M) is the L2 based Sobolev space of any order k∈ ℕ. We investigate how the presence of a characteristic submanifold, namely a submanifold tangent to all vector fields Xj, may cause a failure of exact regularity. We prove that this indeed happens for a class of "worm sublaplacians", which are real analogues of Kohn Laplacians on Diederich--Fornaess worm domains, of interest in several complex variables. Our main tool is a scaling lemma inspired by work of Barrett and Christ on the ∂̅-Neumann problem.
Related
- General regularity obstructions for the arrival time equation
- Autonomous Flows: Exact Finite Interpolation and Uniform Approximation Obstructions
- From Pre-triangulation to Triangulation: Obstructions and Exact Lifting
- Regularity of Manhattan manifolds and exact dimensionality for relatively Anosov groups
- Super-g-Hölder Regularity in Stieltjes Dynamics: Atomic Structure, Exact Extremal Values, and Recovery of the Stieltjes Clock
BibTeX
@misc{dallara2026obstructions,
title = {{Obstructions to exact regularity of sublaplacians}},
author = {Gian Maria Dall'Ara},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03037},
eprint = {2610.03037},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03037},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Obstructions to exact regularity of sublaplacians
AU - Gian Maria Dall'Ara
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03037
UR - https://arxiv.org/abs/2610.03037
AB - Let M be a closed manifold equipped with a collection of smooth real vector fields X_1,…, X_k and a smooth measure μ. Let $P:=\sum_j X_j^\dagger X_j $ be the associated sublaplacian. Under the assumption that each pair of points of M can be connected by a path obtained by joining integral curves of the vector fields, the equation Pu=f admits unique zero-average solutions u in the natural energy space, for all zero average data f∈ L^2. We say that exact regularity holds for the above equation if u∈ H^k(M) whenever f∈ H^k(M), where H^k(M) is the L^2 based Sobolev space of any order k∈ ℕ. We investigate how the presence of a characteristic submanifold, namely a submanifold tangent to all vector fields X_j, may cause a failure of exact regularity. We prove that this indeed happens for a class of "worm sublaplacians", which are real analogues of Kohn Laplacians on Diederich--Fornaess worm domains, of interest in several complex variables. Our main tool is a scaling lemma inspired by work of Barrett and Christ on the ∂̅-Neumann problem.
ER -