Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Explicit formulas for the Szegő kernel on certain weakly pseudoconvex domains

Authors: Gábor Francsics and Nicholas Hanges
Journal: Proc. Amer. Math. Soc. 123 (1995), 3161-3168
MSC: Primary 32H10; Secondary 32F15
MathSciNet review: 1301494
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The objective of this paper is to determine the Szegő kernel of the domain $ \mathcal{D} = \{ (z,\zeta ,w) \in {\mathbb{C}^{n + m + 1}};\Im {\text{m}}w > {\left\Vert z \right\Vert^2} + {\left\Vert \zeta \right\Vert^{2p}}\} $ explicitly in closed form.

References [Enhancements On Off] (What's this?)

  • [C] M. Christ, Remarks on the breakdown of analyticity for $ {\bar \partial _b}$, and Szegő kernels, Harmonic Analysis, ICM-90 Satellite Conference Proceedings, Springer-Verlag, New York, 1991, pp. 61-78. MR 1261429 (94k:32028)
  • [CG] M. Christ and D. Geller, Counterexamples to analytic hypoellipticity for domains of finite type, Ann. of Math. (2) 135 (1992), 551-566. MR 1166644 (93i:35034)
  • [D'A1] J. P. D'Angelo, A note on the Bergman kernel, Duke Math. J. 45 (1978), 259-265. MR 0473231 (57:12906)
  • [D'A2] -, An explicit computation of the Bergman kernel function, J. Geom. Anal. 4 (1994), 23-34. MR 1274136 (95a:32039)
  • [DT] M. Derridj and D. Tartakoff, Local analyticity for the $ \bar \partial $-Neumann problem and $ {\square _b}$ -- Some model domains without maximal estimates, Duke Math. J. 64 (1991), 377-402. MR 1136382 (92j:32055)
  • [FH] G. Francsics and N. Hanges, The Bergman kernel of complex ellipsoids and hypergeometric functions in several variables, preprint, 1994, pp. 1-10.
  • [GS] P. C. Greiner and E. M. Stein, On the solvability of some differential operators of type $ {\square _b}$, Proceedings of International Conferences, Cortona, Italy, 1976-77, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) (1978), 106-165. MR 681306 (84d:35111)
  • [H] N. Hanges, Explicit formulas for the Szegő kernel for some domains in $ {\mathbb{C}^2}$, J. Funct. Anal. 88 (1990), 153-165. MR 1033918 (90k:32016)
  • [Ka] H. Kang, $ {\bar \partial _{b}}$-equations on certain unbounded weakly pseudoconvex domains, Trans. Amer. Math. Soc. 315 (1989), 389-413. MR 989577 (90a:35172)
  • [Ko] A. Korányi, The Poisson integral for generalized half-planes and bounded symmetric domains, Ann. of Math. (2) 82 (1965), 332-350. MR 0200478 (34:371)
  • [M1] M. Machedon, Estimates for the parametrix of the Kohn Laplacian on (0,1) forms on certain weakly pseudoconvex domains, Ph.D. Thesis, Princeton University, Princeton, NJ, 1986.
  • [M2] -, Estimates for the parametrix of the Kohn Laplacian on certain domains, Invent. Math. 91 (1988), 339-364. MR 922804 (89d:58118)
  • [Mc] J. McNeal, Local geometry of decoupled pseudoconvex domains, Aspects of Mathematics, Proceedings of the International Workshop, Wuppertal, 1990, pp. 223-230. MR 1122183 (92g:32033)
  • [N] A. Nagel, Vector fields and nonisotropic metrics, Beijing Lectures in Harmonic Analysis, Ann. of Math. Stud., no. 112, Princeton Univ. Press, Princeton, NJ, 1986, pp. 241-306. MR 864374 (88f:42045)
  • [NRSW] A. Nagel, J. P. Rosay, E. M. Stein, and S. Wainger, Estimates for the Bergman and Szegö kernels in $ {\mathbb{C}^2}$, Ann. of Math. (2) 129 (1989), 113-149. MR 979602 (90g:32028)
  • [S] E. M. Stein, Note on the boundary values of holomorphic functions, Ann. of Math. (2) 82 (1965), 351-353. MR 0188485 (32:5923)
  • [SW] E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Math. Ser., vol. 32, Princeton Univ. Press, Princeton, NJ, 1971. MR 0304972 (46:4102)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 32H10, 32F15

Retrieve articles in all journals with MSC: 32H10, 32F15

Additional Information

Keywords: Szegő kernel, weakly pseudoconvex domains, explicit formulas
Article copyright: © Copyright 1995 American Mathematical Society

American Mathematical Society