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A New Proof of Okaji’s Theorem for a Class of Sum of Squares Operators

Paulo D. Cordaro, Nicholas Hanges (2009)

Annales de l’institut Fourier

Let P be a linear partial differential operator with analytic coefficients. We assume that P is of the form “sum of squares”, satisfying Hörmander’s bracket condition. Let q be a characteristic point for P . We assume that q lies on a symplectic Poisson stratum of codimension two. General results of Okaji show that P is analytic hypoelliptic at q . 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...

A propagation theorem for a class of microfunctions

Andrea D'Agnolo, Giuseppe Zampieri (1990)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Let A be a closed set of M R n , whose conormai cones x + y x * A , x A , have locally empty intersection. We first show in §1 that dist x , A , x M A is a C 1 function. We then represent the n microfunctions of C A | X , X C n , using cohomology groups of O X of degree 1. By the results of § 1-3, we are able to prove in §4 that the sections of C A | X π ˙ - 1 x 0 , x 0 A , satisfy the principle of the analytic continuation in the complex integral manifolds of H ϕ i C i = 1 , , m , ϕ i being a base for the linear hull of γ x 0 * A in T x 0 * M ; in particular we get Γ A × M T * M X C A | X A × M T ˙ * M X = 0 . When A is a half space with C ω -boundary,...

Application of linear hyperbolic PDE to linear quantum fields in curved spacetimes : especially black holes, time machines and a new semi-local vacuum concept

Bernard Kay (2000)

Journées équations aux dérivées partielles

Several situations of physical importance may be modelled by linear quantum fields propagating in fixed spacetime-dependent classical background fields. For example, the quantum Dirac field in a strong and/or time-dependent external electromagnetic field accounts for the creation of electron-positron pairs out of the vacuum. Also, the theory of linear quantum fields propagating on a given background curved spacetime is the appropriate framework for the derivation of black-hole evaporation (Hawking...

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