CR functions vanishing on open sets (almost) complex structures and Cohen's example

Abstract

This paper deals with (non-)uniqueness in the Cauchy problem, for functions annihilated by a complex vector field. (Do Lu ≡ 0 and the Cauchy data u = 0 on some non characteristic hypersurface imply u ≡ 0, in a neighborhood of the hypersurface?) Most of the results are more or less well known. But the proofs/constructions are mostly new, using very elementary considerations on (almost) complex structures. Hopefully this makes the topic more accessible, in particular to complex analysts. The failure of uniqueness in the Cauchy problem provides a striking example of non embeddable strictly pseudoconvex CR structure.