Explicit formulas for the Szegő kernel on certain weakly pseudoconvex domains
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- by Gábor Francsics and Nicholas Hanges PDF
- Proc. Amer. Math. Soc. 123 (1995), 3161-3168 Request permission
Abstract:
The objective of this paper is to determine the Szegő kernel of the domain $\mathcal {D} = \{ (z,\zeta ,w) \in {\mathbb {C}^{n + m + 1}};\Im {\text {m}}w > {\left \| z \right \|^2} + {\left \| \zeta \right \|^{2p}}\}$ explicitly in closed form.References
- Michael Christ, Remarks on the breakdown of analyticity for $\overline \partial _b$ and Szegő kernels, Harmonic analysis (Sendai, 1990) ICM-90 Satell. Conf. Proc., Springer, Tokyo, 1991, pp. 61–78. MR 1261429
- Michael Christ and Daryl Geller, Counterexamples to analytic hypoellipticity for domains of finite type, Ann. of Math. (2) 135 (1992), no. 3, 551–566. MR 1166644, DOI 10.2307/2946576
- John P. D’Angelo, A note on the Bergman kernel, Duke Math. J. 45 (1978), no. 2, 259–265. MR 473231
- John P. D’Angelo, An explicit computation of the Bergman kernel function, J. Geom. Anal. 4 (1994), no. 1, 23–34. MR 1274136, DOI 10.1007/BF02921591
- Makhlouf Derridj and David S. Tartakoff, Local analyticity for the $\overline \partial$-Neumann problem and $\square _b$—some model domains without maximal estimates, Duke Math. J. 64 (1991), no. 2, 377–402. MR 1136382, DOI 10.1215/S0012-7094-91-06419-7 G. Francsics and N. Hanges, The Bergman kernel of complex ellipsoids and hypergeometric functions in several variables, preprint, 1994, pp. 1-10.
- P. C. Greiner and E. M. Stein, On the solvability of some differential operators of type $cm_{b}$, Several complex variables (Cortona, 1976/1977) Scuola Norm. Sup. Pisa, Pisa, 1978, pp. 106–165. MR 681306
- Nicholas Hanges, Explicit formulas for the Szegő kernel for some domains in $\textbf {C}^2$, J. Funct. Anal. 88 (1990), no. 1, 153–165. MR 1033918, DOI 10.1016/0022-1236(90)90123-3
- Hyeonbae Kang, $\overline \partial _b$-equations on certain unbounded weakly pseudo-convex domains, Trans. Amer. Math. Soc. 315 (1989), no. 1, 389–413. MR 989577, DOI 10.1090/S0002-9947-1989-0989577-5
- Adam Korányi, The Poisson integral for generalized half-planes and bounded symmetric domains, Ann. of Math. (2) 82 (1965), 332–350. MR 200478, DOI 10.2307/1970645 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.
- Matei Machedon, Estimates for the parametrix of the Kohn Laplacian on certain domains, Invent. Math. 91 (1988), no. 2, 339–364. MR 922804, DOI 10.1007/BF01389371
- Jeffery D. McNeal, Local geometry of decoupled pseudoconvex domains, Complex analysis (Wuppertal, 1991) Aspects Math., E17, Friedr. Vieweg, Braunschweig, 1991, pp. 223–230. MR 1122183
- Alexander Nagel, Vector fields and nonisotropic metrics, Beijing lectures in harmonic analysis (Beijing, 1984) Ann. of Math. Stud., vol. 112, Princeton Univ. Press, Princeton, NJ, 1986, pp. 241–306. MR 864374
- A. Nagel, J.-P. Rosay, E. M. Stein, and S. Wainger, Estimates for the Bergman and Szegő kernels in $\textbf {C}^2$, Ann. of Math. (2) 129 (1989), no. 1, 113–149. MR 979602, DOI 10.2307/1971487
- E. M. Stein, Note on the boundary values of holomorphic functions, Ann. of Math. (2) 82 (1965), 351–353. MR 188485, DOI 10.2307/1970646
- Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR 0304972
Additional Information
- © Copyright 1995 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 123 (1995), 3161-3168
- MSC: Primary 32H10; Secondary 32F15
- DOI: https://doi.org/10.1090/S0002-9939-1995-1301494-8
- MathSciNet review: 1301494