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We study the top-degree cohomology for the $${\bar{\partial }_b}$$ operator defined on a generic submanifold of the complex space as well as for the differential complex associated with a locally integrable structure $${\mathcal {V}}$$ over a smooth manifold. The main assumptions are that $${\mathcal {V}}$$ is hypocomplex and that the differential complex is locally solvable in degree one. One of the main tools is an adaptation of a sheaf theoretical argument due to Ramis–Ruget–Verdier.

Inspired by the work of Suzuki [12] on the concept of real-analytic solvability for first-order analytic linear partial differential operators we extend his results for the differential complexes associated to analytic locally integrable structures of corank one. We prove that such notion of solvability is related to the smooth solvability condition introduced by F. Treves [13] in 1983. In our arguments the natural extension to closed forms of the well-known Baouendi–Treves approximation formula, the so-called “Approximate Poincare Lemma” (cf. [1] , [14] ), plays a key role.

We obtain a global version in the N-dimensional torus of the Metivier inequality for analytic and Gevrey hypoellipticity, and based on it we introduce a class of globally analytic hypoelliptic operators which remain so after suitable lower order perturbations. We also introduce a new class of analytic (pseudodifferential) operators on the torus whose calculus allows us to study the corresponding perturbation problem in a far more general context.

In this work we prove that for linear partial differential operators in two variables the well known Hormander's semi-global solvability theorem [6] can be improved by showing that solutions can in fact be taken with a sharp loss of one derivative.

We discuss the Borel map at a point p ∈ K ˆ , where K ˆ is the polynomially convex hull of the compact set K ⊂ C , associating to every function f ∈ A ∞ ( K ) its formal Taylor series at p, and show that it satisfies an alternative (provided that K fulfills a mild condition): It is either surjective or injective, and the latter is the case if and only if p ∈ K ˆ ∘ . We also discuss variants of this result for approximation by rational functions and a smooth version of the Mergelyan theorem.

We use the theory of dual of Fréchet-Schwartz (DFS) spaces to establish a sufficient condition for top-degree solvability for the differential complex associated to a hypocomplex locally integrable structure. As an application, we show that the top-degree cohomology of left-invariant hypocomplex structures on a compact Lie group can be computed only by using left-invariant forms, thus reducing the computation to a purely algebraic one. In the case of left-invariant elliptic involutive structures on compact Lie groups, under certain reasonable conditions, we prove that the cohomology associated to the involutive structure can be computed only by using left-invariant forms. Usamos a teoria da espaços duais de Fréchet-Schwartz (DFS) para estabelecer uma condição suficiente para resolubilidade em grau máximo para o complexo associado a estrutuas localmente integráveis hipocomplexas. Como aplicação, provamos que a cohomologia de estruturas hipocomplexas invariantes à esquerda podem ser calculadas usando apenas formas invariantes à esquerda, assim reduzindo o cálculo a um método puramente algébrico. No caso de estruturas invariantes à esquerda, sob certas condições razoáveis, provamos que a cohomologia associada à estrutura pode ser calculada usando apenas formas invariantes à esquerda.

In this work we study the property of strong unique continuation, at a given point, for Gevrey solutions to homogeneous systems of PDE defined by complex, real-analytic vector fields in involution. We show that when the system is minimal at the point then the strong unique continuation property holds for Gevrey solutions of order σ ∈ [ 1 , 2 ] and, furthermore, when the minimality property fails to hold then there are non-trivial Gevrey flat solutions of any given order σ > 1 . The case of Gevrey order σ > 2 is also studied for some particular classes of involutive systems.

In this work we study the Borel property for smooth solutions to systems of complex vector fields associated to locally integrable structures. Inspired by the recent article [6], in which the Borel property was studied for generic submanifolds of the complex space, we prove similar results in this more general set up. In particular we obtain, for the case of corank one structures, a necessary and sufficient condition for the validity of the Borel property.

Soit P un operateur differentiel analytique, de la forme "somme de carres", avec la condition d'Hormander realisee. Soit q un point caracteristique de P. On suppose que q est un point d'un "symplectic Poisson stratum'' de codimension deux (au sens de Treves). D'apres le theoreme d'Okaji, P est hypoelliptique analytique en q. Autrement dit, la conjecture de Treves est vraie en codimension deux. On donne dans ce travail une preuve elementaire de ce fait.

In this work we discuss the problem of smooth and analytic regularity for hyperfunction solutions to linear partial differential equations with analytic coefficients. In particular we show that some well known “sum of squares” operators, which satisfy Hormander’s condition and consequently are hypoelliptic, admit hyperfunction solutions that are not smooth (in particular they are not distributions).

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.

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.

In this work we study the problem of local solvability, in degree one, for the differential complex associated to a real-analytic, involutive structure of corank one. We present a new exposition of the original arguments due to F. Treves [15] and, as a consequence, we derive more precise results by allowing less regularity for the given one closed-forms.

A necessary and sufficient condition is proved for the validity of the Poincare 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.

This work defines the boundary value of a cohomology class of degree q (0 ≤ q ≤ m −1) valued in the sheaf of germs of holomorphic functions, in a wedge whose edge lies on an arbitrary m-dimensional totally real (smooth) submanifold X of Cm and whose directing cone has its singular homology in dimension q generated by one q-cycle c ⊂ Sm − 1. The boundary value is defined as a hyperfunction in X. After "microlocalizing" about the cycle , it is equivalent to use the Dolbeaurt or the Čech realizations of the cohomology. Provided the cone generated by c bounds a convex cone the boundary value map is injective. The corresponding spaces of (germs of) hyperfunctions are characterized by their hypo-analytic wave-front set. Locally, every hyperfunction solution of the wave equation in Rm (m ≥ 3) is the boundary value of a cohomology class of degre m− 2 in cones about deleted hyperplanes that do not intersect the light cone.

In this work we study C∞-hypoellipticity in spaces of ultradistributions for analytic linear partial differential operators. Our main tool is a new a-priori inequality, which is stated in terms of the behaviour of holomorphic functions on appropriate wedges. In particular, for sum of squares operators satisfying Hormander’s condition, we thus obtain a new method for studying analytic hypoellipticity for such a class. We also show how this method can be explicitly applied by studying a model operator, which is constructed as a perturbation of the so-called Baouendi-Goulaouic operator.

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).

Motivated by the celebrated example of Y. Kannai of a linear partial differential operator which is hypoelliptic but not locally solvable, we consider a class of evolution operators with real-analytic coefficients and study their local solvability both in L 2 and in the weak sense. In order to do so we are led to propose a generalization of the Nirenberg–Treves condition (ψ) which is suitable to our study.

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.

Detailing the main methods in the theory of involutive systems of complex vector fields this book examines the major results from the last twenty five years in the subject. One of the key tools of the subject - the Baouendi-Treves approximation theorem - is proved for many function spaces. This in turn is applied to questions in partial differential equations and several complex variables. Many basic problems such as regularity, unique continuation and boundary behaviour of the solutions are explored. The local solvability of systems of partial differential equations is studied in some detail. The book provides a solid background for others new to the field and also contains a treatment of many recent results which will be of interest to researchers in the subject.

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.

Cusp-type singularities of real analytic curves in the complex plane by P. Ahern and X. Gong The F. and M. Riesz property for vector fields by S. Berhanu and J. Hounie Gevrey hypo-ellipticity for sums of squares of vector fields: Some examples by A. Bove $W^{1,1}$-maps with values into $S^1$ by H. Brezis, P. Mironescu, and A. C. Ponce Analytic hypoellipticity and spectral problems for Schrodinger's equation by S. Chanillo Formal solutions for higher order nonlinear totally characteristic PDEs with irregular singularities by H. Chen and Z. Luo Uniqueness of $L^\infty$ solutions for a class of conormal $BV$ vector fields by F. Colombini and N. Lerner Impact of lower order terms on a model pde in two variables by P. D. Cordaro and N. Hanges Global analytic hypoellipticity for a class of quasilinear sums of squares of vector fields by M. Derridj and D. S. Tartakoff Representations via overdetermined systems by M. Eastwood A semi-explicit fundamental domain for a Picard modular group in complex hyperbolic space by G. Francsics and P. D. Lax Complex horospherical transform on real sphere by S. Gindikin On analyticity in space variable of solutions to the KdV equation by J. Gorsky and A. A. Himonas The multinomial distribution and some Bergman kernels by L. Hormander Several results for holomorphic mappings from $B^n$ into $B^N$ by X. Huang, S. Ji, and D. Xu Whitney and Mizohata structures by H. Jacobowitz One-side Liouville theorems for a class of hypoelliptic ultraparabolic equations by A. E. Kogoj and E. Lanconelli Acyclic sheaves in Banach spaces by L. Lempert A Liouville type theorem for some conformally invariant fully nonlinear equations by A. Li and Y. Y. Li Norm closure of classical pseudodifferential operators does not contain Hormander's class by S. T. Melo Remarks on the well-posedness of the nonlinear Cauchy problem by G. Metivier Representation of solutions of planar elliptic vector fields with degeneracies by A. Meziani Some recollections of working with Francois Treves by L. Nirenberg Boundary value problems on Lipschitz domains in $\mathbb{R}^n$ or $\mathbb{C}^n$ by M.-C. Shaw Hyperbolic systems well posed in all Gevrey classes by S. Spagnolo.

We study the local exactness at the top degree level in the differential complex defined by a smooth, locally integrable structure of rank n in [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]. If Z denotes a local first integral of the structure it is proved that the vanishing of the local cohomology in degree n is implied by the absence of compact connected components of the "fibers" Z = const. This adds one more result towards the verification of a conjecture due to F. Treves regarding the vanishing of the local cohomology of such complexes of differential operators.

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.