Unique Continuation for Continuous CR Functions

Let M be an embedded CR manifold, \(p\in \Gamma \subseteq M\) $p\in \mathrm{\Gamma }\subseteq M$ such that \(T_p\Gamma + J(T_p\Gamma )=T_pM+J(T_pM)\) $T_p\mathrm{\Gamma }+J(T_p\mathrm{\Gamma })=T_{p}M+J(T_{p}M)$ . If \(f=u+iv\) $f=u+iv$ is a continuous CR function whose real part u satisfies \(|u(z)|\le a\exp \left( \frac{-b}{|z-p|}\right) \) $|u(z)|\le aexp\left(\frac{-b}{|z-p|}\right)$ for \(z\in \Gamma \) $z\in \mathrm{\Gamma }$ and f vanishes to infinite order at p, then f vanishes on the CR orbit through p. If the exponential decay is satisfied by f itself, the result is due to Joricke (J Geom Anal 6(4):551–611, 1996).