On the vanishing rate of smooth CR functions

Abstract:

Let $M$ be a lineally convex hypersurface of $\mathbb C^n$ of finite type, $0\in M$. Then there exist non-trivial smooth CR functions on $M$ that are flat at $0$, i.e. whose Taylor expansion about $0$ vanishes identically. Our aim is to characterize the rate at which flat CR functions can decrease without vanishing identically. As it turns out, non-trivial CR functions cannot decay arbitrarily fast, and a possible way of expressing the critical rate is by comparison with a suitable exponential of the modulus of a local peak function.