Nodal sets and horizontal singular sets of \(\mathbb{H}\)-harmonic functions on the Heisenberg group (Q2876621)

This paper studies properties of the nodal set (zero set) of a harmonic function \(u\), with respect to the subelliptic sub-Laplacian, on the Heisenberg group \(\mathbb{H}^n\) of dimension \(2n+1\). The authors recall the notion of the frequency of \(u\) on a ball \(B\) under the gauge distance, as introduced in [\textit{N. Garofalo} and \textit{E. Lanconelli}, Ann. Inst. Fourier 40, No. 2, 313--356 (1990; Zbl 0694.22003)]. Roughly speaking, the frequency of \(u\) is \(\int_{\partial B} u^2 w / \int_{B} |\nabla u|^2\), where \(w\) is a weight related to the Green's function, and \(\nabla\) is the horizontal sub-gradient. They then show that the \(2n\)-dimensional Hausdorff measure of the nodal set (zero set) of \(u\) in a ball is bounded above in terms of the frequency of \(u\).NEWLINENEWLINEAn additional result describes the structure of the horizontal singular set of \(u\), which is the set where both \(u\) and \(\nabla u\) vanish: this set is shown to be a countable union of \(C^1\) submanifolds having dimension at most \(2n-1\).