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Geometric and analytic boundary invariants on pseudoconvex domains. Comparison results. (English) Zbl 0786.32016

The authors’ abstract: “We consider for smooth pseudoconvex bounded domains \(\Omega \subset \mathbb{C}^ n\) of finite type as local analytic invariants on the boundary the growth orders of the Bergman kernel and the Bergman metric and the best possible order of subellipticity \(\varepsilon_ 1>0\) for the \(\overline\partial\)-Neumann problem. Furthermore, we consider as local geometric invariants on \(\partial\Omega\) the order of extendability, the exponent of extendability, the 1-type, and the multitype. Various new inequalities between these invariants are proved, giving in particular analytic information from geometric input. On the other hand, a careful consideration of several series of examples of such domains \(\Omega\) shows that starting from \(n \geq 3\) (essentially) each of these invariants is independent of the remaining ones”.

MSC:

32T99 Pseudoconvex domains
32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
32F45 Invariant metrics and pseudodistances in several complex variables
32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators
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[1] D’Angelo, J. P., Real hypersurfaces, orders of contact, and applications, Ann. of Math., 115, 615-637 (1982) · Zbl 0488.32008 · doi:10.2307/2007015
[2] D’Angelo, J. P., Finite-type conditions for real hypersurfaces in ℂ^n, Complex Analysis, Springer Lecture Notes, 1268, 83-102 (1986) · Zbl 0649.32015
[3] Bloom, Th. Remarks on type conditions for real hypersurfaces in ℂ^n.Several Complex Variables, Proc. Intl. Conf. Cortona, pp. 14-24 (1978).
[4] Christ, M., Regularity properties of the \(\bar \partial - equation\) on weakly pseudoconvex CR-manifolds of dimension 3, Journal AMS, 1, 587-646 (1988) · Zbl 0671.35017
[5] Catlin, D., Boundary invariants of pseudoconvex domains, Ann. of Math., 120, 529-586 (1984) · Zbl 0583.32048 · doi:10.2307/1971087
[6] Catlin, D., Estimates of invariant metrics on pseudoconvex domains of dimension two, Math. Z., 200, 429-466 (1989) · Zbl 0661.32030 · doi:10.1007/BF01215657
[7] Catlin, D., Subelliptic estimates for the \(\bar \partial - Neumann\) problem on pseudoconvex domains, Ann. of Math., 126, 131-191 (1987) · Zbl 0627.32013 · doi:10.2307/1971347
[8] Catlin, D., Necessary conditions for subellipticity of the \(\bar \partial - Neumann\) problem, Ann. of Math., 117, 147-171 (1983) · Zbl 0552.32017 · doi:10.2307/2006974
[9] Cho, S., A lower bound on the Kobayashi metric near a point of finite type in ℂ^n, J. Geom. Anal., 2, 4, 317-325 (1992) · Zbl 0756.32015
[10] Diederich, K.; Fornaess, J. E., Proper holomorphic maps into pseudoconvex domains with real-analytic boundary, Ann. of Math., 110, 575-592 (1979) · Zbl 0394.32012 · doi:10.2307/1971240
[11] Diederich, K.; Fornaess, J. E.; Herbort, G., Boundary behavior of the Bergman metric, Proc. of Symp. in Pure Math., 41, 59-67 (1984) · Zbl 0533.32012
[12] Diederich, K.; Herbort, G.; Ohsawa, T., The Bergman kernel on uniformly extendable pseudoconvex domains, Math. Ann., 273, 471-478 (1986) · Zbl 0582.32028 · doi:10.1007/BF01450734
[13] Diederich, K., and Lieb, I. Konvexitaet in der komplexen Analysis. InDMV Seminar, vol. 2. Birkhaeuser Basel, 1981. · Zbl 0473.32015
[14] Fefferman, Ch., The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Inv. Math., 26, 1-65 (1974) · Zbl 0289.32012 · doi:10.1007/BF01406845
[15] Fefferman, Ch., Parabolic invariant theory in complex analysis, Adv. in Math., 31, 131-262 (1979) · Zbl 0444.32013 · doi:10.1016/0001-8708(79)90025-2
[16] Fornaess, J. E.; Sibony, N., Construction of plurisubharmonic functions on weakly pseudoconvex domains, Duke Math. J., 58, 633-655 (1989) · Zbl 0679.32017 · doi:10.1215/S0012-7094-89-05830-4
[17] Herbort, G. Wachstumsordnung des Bergmankerns auf pseudokonvexen Gebieten.Schriftenreihe des Mathematischen Instituts der Universitaet Muenster, 2 Serie,46 (1987). · Zbl 0636.32013
[18] Herbort, G., Logarithmic growth of the Bergman kernel for weakly pseudoconvex domains in ω^3 of finite type, Manuscr. Math., 45, 69-76 (1983) · Zbl 0559.32006 · doi:10.1007/BF01168581
[19] Kohn, J. J., Subellipticity for the \(\bar \partial - Neumann\) problem on pseudoconvex domains: Sufficient conditions, Acta Math., 142, 79-122 (1979) · Zbl 0395.35069 · doi:10.1007/BF02395058
[20] Kohn, J. J., Boundary behavior of \(\bar \partial\) on weakly pseudoconvex manifolds of dimension two, J. Diff. Geom., 6, 523-543 (1972) · Zbl 0256.35060
[21] Kohn, J. J.; Fefferman, Ch., Hölder estimates on domains of complex dimension two and on three-dimensional CR-manifolds, Advances in Math., 69, 223-303 (1988) · Zbl 0649.35068 · doi:10.1016/0001-8708(88)90002-3
[22] McNeal, J., Boundary behavior of the Bergman kernel function in ℂ^2, Duke Math. J., 58, 499-512 (1989) · Zbl 0675.32020 · doi:10.1215/S0012-7094-89-05822-5
[23] McNeal, J., Lower bounds on the Bergman metric near a point of finite type, Ann. Math., 136, 339-360 (1992) · Zbl 0764.32006 · doi:10.2307/2946608
[24] Nagel, A.; Rosay, J. P.; Stein, E. M.; Wainger, St., Estimates for the Bergman and Szegö kernels in ℂ^2, Ann. Math., 129, 113-149 (1989) · Zbl 0667.32016 · doi:10.2307/1971487
[25] Ohsawa, T., Boundary behavior of the Bergman kernel function on pseudoconvex domains, Publ. RIMS, 20, 897-902 (1984) · Zbl 0569.32013 · doi:10.2977/prims/1195180870
[26] Ohsawa, T.; Takegoshi, K., Extension of L^2-holomorphic functions, Math. Z., 195, 197-204 (1987) · Zbl 0625.32011 · doi:10.1007/BF01166457
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