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Analytic regularity for the Bergman kernel

Gabor Françis;  Nicholas Hanges — 1998

Journées équations aux dérivées partielles

Let Ω ⊂ ℝ 2 be a bounded, convex and open set with real analytic boundary. Let T Ω ⊂ ℂ 2 be the tube with base Ω , and let ℬ be the Bergman kernel of T Ω . If Ω is strongly convex, then ℬ is analytic away from the boundary diagonal. In the weakly convex case this is no longer true. In this situation, we relate the off diagonal points where analyticity fails to the Trèves curves. These curves are symplectic invariants which are determined by the CR structure of the boundary of T Ω . Note that Trèves curves exist only...

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...

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