Paulo D. Cordaro

Global Solvability for a Class of Complex Vector Fields on the Two-Torus

Abstract We consider complex vector fields L on the two-torus. We regard L as an operator acting on smooth functions and study conditions for L to have a closed range. We also give conditions for the range of L to have finite codimension. Our results involve condition (P) of Nirenberg and Treves. One-dimensional orbits diffeomorphic to the unit circle are allowed.

GLOBAL ANALYTIC REGULARITY FOR SUMS OF SQUARES OF VECTOR FIELDS

We consider a class of operators in the form of a sum of squares of vector fields with real analytic coefficients on the torus and we show that the zero order term may influence their global analytic hypoellipticity. Also we extend a result of Cordaro-Himonas.

Homology and cohomology in hypo-analytic structures of the hypersurface type

The work is concerned with local exactness in the cohomology of the differential complex associated with a hypo-analytic structure on a smooth manifold. Only structures of the hypersurface type are considered, i.e., structures in which the rank of the characteristic set does not exceed one. Among them are the CR structures of real hypersurfaces in a complex manifold. The main theorem states anecessary condition for local exactness in dimensionq to hold. The condition is stated in terms of the natural homology associated with the differential complex, as inherited by the level sets of the imaginary part of an arbitrary solutionw whose differential spans the characteristic set at the central point. An intersection number, which generalizes the standard number in singular homology, is defined; the condition is that this number, applied to the intersection of the level sets ofImw with the hypersurfaceRew=0, vanish identically. In a CR structure, and in top dimension, this is shown to be equivalent to the property that the Levi form not be definite at any point—a property, that is likely to be also sufficient for local solvability.

Normalization of complex-valued planar vector fields which degenerate along a real curve

Abstract Taking as a start point the recent article of Meziani [7], we present several results concerning the normalization of a class of complex vector fields in the plane which degenerate along a real curve. We mainly deal with operators with finite regularity and analyze both the local situation as well as the case of normalization near a circle. Some related questions (e.g., on semi-global solvability and on the normalization of a class of generalized Mizohata operators) are also discussed.

Local solvability for a class of differential complexes

In 1983 F. Treves IT1] initiated the study of the local solvability for the class of differential complexes defined by a smooth, locally integrable structure of rank n in a n + l . If Z denotes a local first integral of the structure, Treves conjectured that the vanishing of the local cohomology in degree q of such a differential complex would be related to the vanishing of the singular homology of the sets Z = const, in dimension q 1 . It is the purpose of this article to complete the proof of this conjecture in its full generality.

Necessary and sufficient conditions for the local solvability in hyperfunctions of a class of systems of complex vector fields.

SummaryA necessary and sufficient condition is proved for the validity of the Poincaré Lemma in degreeq≧1, in the differential complex attached to a locally integrable structure of codimension one, in spaces of hyperfunctions. The base manifold ℳ is only assumed to be smooth. The hyperfunctions are defined in the hypo-analytic structure associated to a smooth “first integral” Z. The condition is that the singular homology of the fibres of the map Z be trivial in dimensionq-1. By the approximation formula of [BT]|the germ of this fibration at a point is independent of the choice of the first integral Z.

Gevrey solvability and Gevrey regularity in differential complexes associated to locally integrable structures

In this work we study some properties of the differential complex associated to a locally integrable (involutive) structure acting on forms with Gevrey coefficients. Among other results we prove that, for such complexes, Gevrey solvability follows from smooth solvability under the sole assumption of a regularity condition. As a consequence we obtain the proof of the Gevrey solvability for a first order linear PDE with real-analytic coefficients satisfying the Nirenberg-Treves condition (P).

Symplectic strata and analytic hypoellipticity

We review various classical results on analytic hypoellipticity for operators with double characteristics. Several examples will be discussed to motivate Treves’ conjecture. Finally we announce regularity results obtained recently.