Smooth and Gevrey Microlocal Hypoellipticity for a Class of Hypocomplex Tube Structures

Abstract

We prove a smooth and Gevrey-$s$ microlocal hypoellipticity result for a system of complex vector fields associated with a real-analytic locally integrable structure of tube type, that is also microlocal hypocomplex. In order to so, we employ the use of a certain partial F.B.I. transform adapted to the locally integrable structure, first introduced by M. S. Baouendi, C. H. Chang and F. Treves, and we prove a microlocal characterization of the smooth and Gevrey-$s$ wave front set in terms of the decay of this partial F.B.I. transform.