Analytic regularity for the Bergman kernel
Journées équations aux dérivées partielles (1998), article no. 5, 11 p.

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.

@article{JEDP_1998____A5_0,
     author = {Fran\c{c}is, Gabor and Hanges, Nicholas},
     title = {Analytic regularity for the {Bergman} kernel},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     eid = {5},
     pages = {1--11},
     publisher = {Universit\'e de Nantes},
     year = {1998},
     zbl = {01808715},
     language = {en},
     url = {http://www.numdam.org/item/JEDP_1998____A5_0/}
}
TY  - JOUR
AU  - Françis, Gabor
AU  - Hanges, Nicholas
TI  - Analytic regularity for the Bergman kernel
JO  - Journées équations aux dérivées partielles
PY  - 1998
SP  - 1
EP  - 11
PB  - Université de Nantes
UR  - http://www.numdam.org/item/JEDP_1998____A5_0/
LA  - en
ID  - JEDP_1998____A5_0
ER  - 
%0 Journal Article
%A Françis, Gabor
%A Hanges, Nicholas
%T Analytic regularity for the Bergman kernel
%J Journées équations aux dérivées partielles
%D 1998
%P 1-11
%I Université de Nantes
%U http://www.numdam.org/item/JEDP_1998____A5_0/
%G en
%F JEDP_1998____A5_0
Françis, Gabor; Hanges, Nicholas. Analytic regularity for the Bergman kernel. Journées équations aux dérivées partielles (1998), article  no. 5, 11 p. http://www.numdam.org/item/JEDP_1998____A5_0/

[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). | MR | Zbl

[2] S.C. Chen, Real analytic regularity of the Szegő projection on circular domains, Pacific J. Math. 148 (1991), pp. 225-235. | MR | Zbl

[3] M. Christ, A necessary condition for analytic hypoellipticity, Mathematical Research Letters, 1, pp. 241-248, (1994). | MR | Zbl

[4] M. Christ, The Szegő projection need not preserve global analyticity, Annals of Math. 143 (1990), pp. 301-330. | MR | Zbl

[5] M. Christ and D. Geller, Counterexamples to analytic hypoellipticity for domains of finite type, Ann. of Math. 135 (1992), pp. 551-566. | MR | Zbl

[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). | MR | Zbl

[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. | MR | Zbl

[8] J. Faraut and A. Koranyi, Analysis on symmetric cones, Oxford University Press, (1994). | MR | Zbl

[9] G. Francsics and N. Hanges, Analytic singularities, Contemporary Mathematics, 205 (1997), pp. 69-78. | MR | Zbl

[10] G. Francsics and N. Hanges, Trèves curves and the Szegő kernel, Indiana University Mathematics Journal, to appear. | Zbl

[11] D. Geller, Analytic pseudodifferential operators for the Heisenberg group and local solvability, Mathematical Notes 37, Princeton University Press (1990). | MR | Zbl

[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. | Zbl

[13] N. Hanges and A. A. Himonas, Analytic hypoellipticity for generalized Baouendi - Goulaouic operators, Journal of Functional Analysis, 125 (1) (1994), pp. 309-325. | MR | Zbl

[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. | MR | Zbl

[15] L. Hörmander, Notions of convexity, Birkhäuser, 1994. | Zbl

[16] L. Hörmander, L2 Estimates and Existence Theorems for the ∂ operator, Acta. Math. 113 (1965), pp. 89-152. | Zbl

[17] M. Kashiwara, Analyse Micro-locale du noyau de Bergman, Séminaire Goulaouic-Schwartz 1976-1977, Exposé VIII. | EuDML | Numdam | Zbl

[18] A. Koranyi, The Bergman kernel function for tubes over convex cones, Pacific J. Math. 12 pp. 1355-1359. | MR | Zbl

[19] S. Krantz, Function theory of several complex variables, John Wiley, 1982. | MR | Zbl

[20] G. Métivier, Une classe d'opérateurs non hypoelliptiques analytiques, Indiana Univ. Math. J. 29 (1980), pp. 823-860. | MR | Zbl

[21] G. Pólya, On the zeros of an integral function represented by Fourier's integral, Messenger of Math., 52 (1923), 185-88. | JFM

[22] G. Pólya, Graeffe's method for eigenvalues, Numerische Mathematik, 11 (1968) 315-319. | EuDML | MR | Zbl

[23] J. Sjöstrand, Analytic wavefront sets and operators with multiple characteristics, Hokkaido Mathematical Journal, 12 (1983) pp. 392-433. | MR | Zbl

[24] D. Tartakoff, Gevrey and analytic hypoellipticity, Microlocal Analysis and Spectral Theory, Kluwer Academic Publishers, L. Rodino, ed. (1997) pp. 39-59. | MR | Zbl

[25] D. Tartakoff, On the Local Real Analyticity of Solutions to ʩb and the ∂ Neumann Problem, Acta. Math. 145 (1980) pp. 117-204. | MR | Zbl

[26] J-M. Trepreau, Sur l'hypoellipticite analytique microlocale des operateurs du type principal, Comm. PDE, 9 (11) (1984), pp. 1119-1146. | MR | Zbl

[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. | MR | Zbl

[28] F. Trèves, Symplectic geometry and analytic hypo-ellipticity, preprint. | Zbl

[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. | MR | Zbl