Uniqueness properties of CR-functions

Abstract:

Let M be a real infinitely differentiable closed hypersurface in X, a complex manifold of complex dimension $n \geqslant 2$. The uniqueness properties of solutions to the system ${\bar \partial _M}u = f$, where ${\bar \partial _M}$ is the induced Cauchy-Riemann operator on M, are of interest in the fields of several complex variables and partial differential equations. Since dM is linear, the study of the solution to the equation ${\bar \partial _M}u = 0$ is sufficient for uniqueness. A ${C^\infty }$ solution to this homogeneous equation is called a CR-function on M. The main result of this article is that a CR-function is uniquely determined, at least locally, by its values on a real k-dimensional ${C^\infty }$ generic submanifold ${S^k}$ of M with $k \geqslant n$. The facts that ${S^k}$ is generic and $k \geqslant n$ together form the lower dimensional analogue of the concept of noncharacteristic.