Unique continuation for continuous CR functions (Q6661036)

The main objective of this paper is the unique continuation of the continuous CR functions defined on smooth CR manifolds embedded in complex spaces. Given a point \(p\) of a CR manifold \(M\), let \(\Omega(M, p)\) be the CR orbit of all points in \(M\) jointed to \(p\) through some piecewise CR curve.\N\NLet \(M\) be a smooth CR submanifold in \(\mathbb C^n\) and consider \(\Gamma\) as a closed submanifold of \(M\) enjoying\N\[\NT_p\Gamma+J(T_p)\Gamma =T_pM+J(T_pM)\N\]\Nwith \(p\in \Gamma\) and with \(J\) being the complex structure of \(M\). Assume that \(f=u+ i v\) is a continuous CR function on \(M\) where its real part satisfies the estimate\N\[\N|u(z)|\leq a\, \exp\Big(\frac{-b}{|z-p|}\Big)\N\]\Nfor every \(a, b >0\) and for every \(z\in\Gamma\). Then, as the main result of this paper, it is proved that if \(f\) vanishes to infinite order at \(p\) then it vanishes as well on the entire orbit \(\Omega(M, p)\). In the specific case where the above exponential estimation is satisfied by \(f\), instead of \(u\), the result was proved by \textit{B. Jöricke} [J. Geom. Anal. 6, No. 4, 555--611 (1996; Zbl 0917.32007)].