# Analytic regularity for the Bergman kernel

• page 1-11
• ISSN: 0752-0360

top

## Abstract

top
Let $\Omega \subset {ℝ}^{2}$ be a bounded, convex and open set with real analytic boundary. Let ${T}_{\Omega }\subset {ℂ}^{2}$ be the tube with base $\Omega ,$ and let $ℬ$ be the Bergman kernel of ${T}_{\Omega }$. If $\Omega$ 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 ${T}_{\Omega }$. Note that Trèves curves exist only when $\Omega$ has at least one weakly convex boundary point.

## How to cite

top

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

top
1. [1] L. Boutet de Monvel, Singularity of the Bergman kernel, Complex Geometry, Lecture Notes in Pure and Applied Mathematics, Vol. 143, Marcel Dekker, Inc. (1993). Zbl0798.32024MR93k:32047
2. [2] S.C. Chen, Real analytic regularity of the Szegő projection on circular domains, Pacific J. Math. 148 (1991), pp. 225-235. Zbl0729.32007MR91m:32023
3. [3] M. Christ, A necessary condition for analytic hypoellipticity, Mathematical Research Letters, 1, pp. 241-248, (1994). Zbl0841.35026MR94m:35068
4. [4] M. Christ, The Szegő projection need not preserve global analyticity, Annals of Math. 143 (1990), pp. 301-330. Zbl0851.32024MR97e:32027
5. [5] M. Christ and D. Geller, Counterexamples to analytic hypoellipticity for domains of finite type, Ann. of Math. 135 (1992), pp. 551-566. Zbl0758.35024MR93i:35034
6. [6] M. Derridj, Analyticité globale de la solution canonique de ∂b pour une classe d'hypersurfaces compactes pseudoconvexes de ℂ², Mathematical Research Letters, 4, pp. 667-677, (1997). Zbl0916.35020MR98j:32016
7. [7] M. Derridj and D. Tartakoff, Microlocal analyticity for the canonical solution to ∂b on some rigid weakly pseudoconvex hypersurfaces in ℂ², Comm. PDE 20 (1995), pp. 1647-1667. Zbl0833.35100MR97e:35121
8. [8] J. Faraut and A. Koranyi, Analysis on symmetric cones, Oxford University Press, (1994). Zbl0841.43002MR98g:17031
9. [9] G. Francsics and N. Hanges, Analytic singularities, Contemporary Mathematics, 205 (1997), pp. 69-78. Zbl0892.32019MR98i:32036
10. [10] G. Francsics and N. Hanges, Trèves curves and the Szegő kernel, Indiana University Mathematics Journal, to appear. Zbl0939.32001
11. [11] D. Geller, Analytic pseudodifferential operators for the Heisenberg group and local solvability, Mathematical Notes 37, Princeton University Press (1990). Zbl0695.47051MR91d:58243
12. [12] A. Grigis and J. Sjöstrand, Front d'onde analytique et sommes de carres de champs de vecteurs, Duke Math. J. 52 (1985), pp. 35-51. Zbl0581.35009
13. [13] N. Hanges and A. A. Himonas, Analytic hypoellipticity for generalized Baouendi - Goulaouic operators, Journal of Functional Analysis, 125 (1) (1994), pp. 309-325. Zbl0812.35026MR95j:35049
14. [14] B. Helffer, Conditions nécessaires d'hypoanalyticité pour des opérateurs invariants à gauche homogènes sur un groupe nilpotent gradué, J. Diff. Eq. 44 (1982), pp. 460-481. Zbl0458.35019MR84c:35026
15. [15] L. Hörmander, Notions of convexity, Birkhäuser, 1994. Zbl0835.32001
16. [16] L. Hörmander, L2 Estimates and Existence Theorems for the ∂ operator, Acta. Math. 113 (1965), pp. 89-152. Zbl0158.11002
17. [17] M. Kashiwara, Analyse Micro-locale du noyau de Bergman, Séminaire Goulaouic-Schwartz 1976-1977, Exposé VIII. Zbl0445.32020
18. [18] A. Koranyi, The Bergman kernel function for tubes over convex cones, Pacific J. Math. 12 pp. 1355-1359. Zbl0114.04002MR27 #1623
19. [19] S. Krantz, Function theory of several complex variables, John Wiley, 1982. Zbl0471.32008MR84c:32001
20. [20] G. Métivier, Une classe d'opérateurs non hypoelliptiques analytiques, Indiana Univ. Math. J. 29 (1980), pp. 823-860. Zbl0455.35041MR82a:35029
21. [21] G. Pólya, On the zeros of an integral function represented by Fourier's integral, Messenger of Math., 52 (1923), 185-88. JFM49.0219.02
22. [22] G. Pólya, Graeffe's method for eigenvalues, Numerische Mathematik, 11 (1968) 315-319. Zbl0191.15903MR37 #2434
23. [23] J. Sjöstrand, Analytic wavefront sets and operators with multiple characteristics, Hokkaido Mathematical Journal, 12 (1983) pp. 392-433. Zbl0531.35022MR85e:35022
24. [24] D. Tartakoff, Gevrey and analytic hypoellipticity, Microlocal Analysis and Spectral Theory, Kluwer Academic Publishers, L. Rodino, ed. (1997) pp. 39-59. Zbl0879.35167MR98e:35038
25. [25] D. Tartakoff, On the Local Real Analyticity of Solutions to ʩb and the ∂ Neumann Problem, Acta. Math. 145 (1980) pp. 117-204. Zbl0456.35019MR81k:35033
26. [26] J-M. Trepreau, Sur l'hypoellipticite analytique microlocale des operateurs du type principal, Comm. PDE, 9 (11) (1984), pp. 1119-1146. Zbl0566.35027MR86m:58144
27. [27] F. Trèves, Analytic hypoellipticity of a class of pseudodifferential operators with double characteristics and applications to the ∂ - Neumann problem, Communications in PDE 3 (1978), pp. 475-642. Zbl0384.35055MR58 #11867
28. [28] F. Trèves, Symplectic geometry and analytic hypo-ellipticity, preprint. Zbl0938.35038
29. [29] E.B. Vinberg, The theory of convex homogeneous cones, Trudy Moscov. Mat. Obsc. 12 pp. 303-358 ; Trans. Moscow Math. Soc. 12 pp. 303-358. Zbl0138.43301MR28 #1637

top

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.