A strong unique continuation result for the Baouendi operator

TL;DR

This work establishes a strong unique continuation property for zero-order perturbations of the degenerate Baouendi operator $\mathscr{B}_\alpha = \Delta_z + |z|^{2\alpha}\Delta_t$, under a Hardy-type growth condition on the perturbation $V$. The authors develop $L^2$ Carleman estimates tailored to the degenerate geometry and combine them with the classical Hardy inequality to control singular perturbations, avoiding frequency-function methods. They prove SUCP when $V$ satisfies $|V| \le \frac{C_0}{\rho_\alpha^{2-\delta}}$, covering both locally bounded and singular potentials, and extend the result to variable-coefficient operators with intrinsic Lipschitz regularity; this requires $m\ge3$ and $0<\alpha\le1$. The approach yields a robust, elementary framework for strong unique continuation on degenerate, subelliptic operators and connects to Heisenberg-type geometries by leveraging the intrinsic gauge and fundamental solutions.

Abstract

We establish a strong unique continuation property for the subelliptic Baouendi operator under the presence of zero-order perturbations satisfying an almost Hardy-type growth condition. In particular, the admissible class includes both $L^\infty_{\mathrm{loc}}$ and singular potentials. We prove that any solution vanishing to infinite order at a point of the degeneracy manifold of the operator must be identically zero. The result holds extends to variable-coefficient operators with intrinsic Lipschitz regularity. A notable feature of the proof is that it relies exclusively on $L^2$ Carleman estimates combined with the classical Hardy inequality.

Theorems & Definitions (10)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Lemma 2.2
• proof
• Theorem 2.3
• Theorem 2.4
• Lemma 2.5
• proof : Proof of Theorem \ref{['main1']}
• proof : Proof of Theorem \ref{['T:La']}