发表机构
Dipartimento di Matematica “T. Levi-Civita”; Scuola Normale Superiore; Scuola Galileiana di Studi Superiori（数学系“T. Levi-Civita”; 比萨高等师范学院; 伽利略高等研究学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在Carnot群中建立满足左不变PDE约束的测度结构定理，引入次椭圆波锥，并利用旋度型算子将Rank-one定理推广至所有Carnot群中的$BV_H$函数。
AI 中文摘要
我们建立了满足Carnot群中左不变偏微分方程约束的测度的结构定理。我们的推广受第一作者与F. Rindler的论文arXiv:1601.06543启发，需要引入次椭圆波锥，这是补偿紧致理论中波锥的次黎曼版本，并利用了齐次空间上调和分析的经典结果。然后，我们利用这一结果将水平有界变差函数（$BV_H$）的Rank-one定理推广到所有Carnot群。这是通过考虑一个合适的旋度型左不变算子来实现的，该算子为水平梯度提供必要的微分约束，并研究这些约束的次椭圆性。
英文摘要
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.
Comments29 pages
