For a class of degenerate pseudodifferential operators, local parametrices are constructed. This is done in the framework of a pseudodifferential calculus upon adding conditions of trace and potential type, respectively, along the boundary on which the operators degenerate.
We give a characterization of constant coefficients elliptic operators in terms of estimates of their iterations on smooth functions.
Let be a linear partial differential operator with analytic coefficients. We assume that is of the form “sum of squares”, satisfying Hörmander’s bracket condition. Let be a characteristic point for . We assume that lies on a symplectic Poisson stratum of codimension two. General results of Okaji show that is analytic hypoelliptic at . Hence Okaji has established the validity of Treves’ conjecture in the codimension two case. Our goal here is to give a simple, self-contained proof of...
Let P be a hypoelliptic polynomial. We consider classes of ultradifferentiable functions with respect to the iterates of the partial differential operator P(D) and prove that such classes satisfy a Paley-Wiener type theorem. These classes and the corresponding test spaces are nuclear.
On montre dans cet article comment des théorèmes récents d’hypoellipticité ou de propagation des singularités peuvent être améliorés par une méthode d’addition de variables qui permet dans certains cas de “désingulariser” l’ensemble caractéristique.
We discuss the open problem of analytic hypoellipticity for sums of squares of vector fields, including some recent partial results and a conjecture of Treves.