Analytic regularity for the Bergman kernel

Gabor Françis; Nicholas Hanges

Journées équations aux dérivées partielles (1998)

  • page 1-11
  • ISSN: 0752-0360

Abstract

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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 when Ω has at least one weakly convex boundary point.

How to cite

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Françis, Gabor, and Hanges, Nicholas. "Analytic regularity for the Bergman kernel." Journées équations aux dérivées partielles (1998): 1-11. <http://eudml.org/doc/93362>.

@article{Françis1998,
abstract = {Let $\Omega \subset \mathbb \{R\}^2$ be a bounded, convex and open set with real analytic boundary. Let $T_\{\Omega \} \subset \mathbb \{C\}^2$ be the tube with base $\Omega ,$ and let $\mathcal \{B\}$ be the Bergman kernel of $T_\{\Omega \}$. If $\Omega $ is strongly convex, then $\mathcal \{B\}$ 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 $T_\{\Omega \}$. Note that Trèves curves exist only when $\Omega $ has at least one weakly convex boundary point.},
author = {Françis, Gabor, Hanges, Nicholas},
journal = {Journées équations aux dérivées partielles},
keywords = {analytic regularity; weakly pseudoconvex domain; analytic-wave front set; Bergman kernel},
language = {eng},
pages = {1-11},
publisher = {Université de Nantes},
title = {Analytic regularity for the Bergman kernel},
url = {http://eudml.org/doc/93362},
year = {1998},
}

TY - JOUR
AU - Françis, Gabor
AU - Hanges, Nicholas
TI - Analytic regularity for the Bergman kernel
JO - Journées équations aux dérivées partielles
PY - 1998
PB - Université de Nantes
SP - 1
EP - 11
AB - Let $\Omega \subset \mathbb {R}^2$ be a bounded, convex and open set with real analytic boundary. Let $T_{\Omega } \subset \mathbb {C}^2$ be the tube with base $\Omega ,$ and let $\mathcal {B}$ be the Bergman kernel of $T_{\Omega }$. If $\Omega $ is strongly convex, then $\mathcal {B}$ 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 $T_{\Omega }$. Note that Trèves curves exist only when $\Omega $ has at least one weakly convex boundary point.
LA - eng
KW - analytic regularity; weakly pseudoconvex domain; analytic-wave front set; Bergman kernel
UR - http://eudml.org/doc/93362
ER -

References

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