Analyticity in the boundary of a pseudoconvex domain

Abstract:

Let $D$ be a bounded pseudoconvex domain with ${C^\infty }$ boundary in ${{\mathbf {C}}^n},{A^\infty }(D)$ the algebra of functions holomorphic in $D$ and ${C^\infty }$ up to the boundary, and $M$ a compact real-analytic manifold in the boundary which is integral for the complex structure of the boundary and which has no complex tangent vectors. A necessary and sufficient condition that each element of ${A^\infty }(D)$ be real-analytic on $M$ is that the germ of the complexification of $M$ be in the boundary. Examples indicate that the quasi-analyticity of ${A^\infty }(D)$ along $M$ is possible even in the absence of complex manifolds in the boundary.