Weak pseudoconcavity and the maximum modulus principle

Abstract

In this paper we focus on the maximum modulus principle and weak unique continuation for CR functions on an abstract almost CR manifold M. It is known that some assumption must be made on M in order to have either of these: it suffices to consider the standard CR structure on the sphere S3 in C2 to see that the maximum modulus principle is not valid in the presence of strict pseudoconvexity. For weak unique continuation, Rosay [R] has shown by an example that there is a strictly pseudoconvex CR structure on R3, which is a perturbation of the aforementioned standard CR structure on S3, such that there exists a smooth CR function u, u ≡ 0, with u ≡ 0 on a nonempty open set. However positive results were obtained in [DCN] under the assumption of pseudoconcavity and in [HN] under the assumption of essential pseudoconcavity (and also the finite kind for the maximum modulus principle). Here we investigate these matters under the assumption of weak pseudoconcavity on M, which is a more general notion than that of essential pseudoconcavity, insofar as it drops the minimality (and the finite kind) hypothesis on M. We obtain sharp results involving propagation along Sussmann leaves. The core of our argument is that on a weakly pseudoconcave M the square of the modulus of a CR function is subharmonic with respect to a degenerate-elliptic operator P on M. We employ a maximum principle for real-valued functions which is in the spirit of [Hf], [Ni], [B], [H]. In order to understand our motivation in considering the weak pseudoconcavity condition on M, the reader is referred to the examples in [HN].