Analysis & PDE

  • Anal. PDE
  • Volume 10, Number 6 (2017), 1429-1454.

Bergman kernel and hyperconvexity index

Bo-Yong Chen

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Abstract

Let Ω n be a bounded domain with the hyperconvexity index α(Ω) > 0. Let ϱ be the relative extremal function of a fixed closed ball in Ω, and set μ := |ϱ|(1 + |log|ϱ||)1 and ν := |ϱ|(1 + |log|ϱ||)n. We obtain the following estimates for the Bergman kernel. (1) For every 0 < α < α(Ω) and 2 p < 2 + 2α(Ω)(2n α(Ω)), there exists a constant C > 0 such that Ω|KΩ( ,w)KΩ (w)|p C|μ(w)|(p2)nα for all w Ω. (2) For every 0 < r < 1, there exists a constant C > 0 such that |KΩ(z,w)|2(KΩ(z)KΩ(w)) C(min{ν(z)μ(w),ν(w)μ(z)})r for all z,w Ω. Various applications of these estimates are given.

Article information

Source
Anal. PDE, Volume 10, Number 6 (2017), 1429-1454.

Dates
Received: 11 November 2016
Revised: 27 February 2017
Accepted: 24 April 2017
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1510843528

Digital Object Identifier
doi:10.2140/apde.2017.10.1429

Mathematical Reviews number (MathSciNet)
MR3678493

Zentralblatt MATH identifier
1368.32003

Subjects
Primary: 32A25: Integral representations; canonical kernels (Szego, Bergman, etc.)
Secondary: 32U35: Pluricomplex Green functions

Keywords
Bergman kernel hyperconvexity index

Citation

Chen, Bo-Yong. Bergman kernel and hyperconvexity index. Anal. PDE 10 (2017), no. 6, 1429--1454. doi:10.2140/apde.2017.10.1429. https://projecteuclid.org/euclid.apde/1510843528


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References

  • M. Adachi and J. Brinkschulte, “A global estimate for the Diederich–Fornaess index of weakly pseudoconvex domains”, Nagoya Math. J. 220 (2015), 67–80.
  • V. V. Andrievskii, “On sparse sets with Green function of the highest smoothness”, Comput. Methods Funct. Theory 5:2 (2005), 301–322.
  • D. E. Barrett, “Irregularity of the Bergman projection on a smooth bounded domain in ${\bf C}\sp{2}$”, Ann. of Math. $(2)$ 119:2 (1984), 431–436.
  • D. E. Barrett, “Behavior of the Bergman projection on the Diederich–Fornæss worm”, Acta Math. 168:1–2 (1992), 1–10.
  • M. Berger, “Encounter with a geometer: Eugenio Calabi”, pp. 20–60 in Manifolds and geometry (Pisa, 1993), edited by P. de Bartolomeis et al., Sympos. Math. 36, Cambridge University, 1996.
  • B. Berndtsson and P. Charpentier, “A Sobolev mapping property of the Bergman kernel”, Math. Z. 235:1 (2000), 1–10.
  • D. Bertilsson, “Coefficient estimates for negative powers of the derivative of univalent functions”, Ark. Mat. 36:2 (1998), 255–273.
  • Z. Błocki, “Estimates for the complex Monge–Ampère operator”, Bull. Polish Acad. Sci. Math. 41:2 (1993), 151–157.
  • Z. Błocki, “The complex Monge–Ampère operator in pluripotential theory”, lecture notes, Jagiellonian University, Kraków, 2002, hook http://gamma.im.uj.edu.pl/~blocki/publ/ln/wykl.pdf \posturlhook.
  • Z. Błocki, “The Bergman metric and the pluricomplex Green function”, Trans. Amer. Math. Soc. 357:7 (2005), 2613–2625.
  • Z. Błocki and P. Pflug, “Hyperconvexity and Bergman completeness”, Nagoya Math. J. 151 (1998), 221–225.
  • H. P. Boas and E. J. Straube, “Sobolev estimates for the $\bar\partial$-Neumann operator on domains in ${\bf C}^n$ admitting a defining function that is plurisubharmonic on the boundary”, Math. Z. 206:1 (1991), 81–88.
  • J. E. Brennan, “The integrability of the derivative in conformal mapping”, J. London Math. Soc. $(2)$ 18:2 (1978), 261–272.
  • E. Calabi, “Isometric imbedding of complex manifolds”, Ann. of Math. $(2)$ 58 (1953), 1–23.
  • L. Carleson, Selected problems on exceptional sets, Van Nostrand Math. Stud. 13, Van Nostrand, Princeton, 1967.
  • L. Carleson and T. W. Gamelin, Complex dynamics, Springer, New York, 1993.
  • L. Carleson and P. W. Jones, “On coefficient problems for univalent functions and conformal dimension”, Duke Math. J. 66:2 (1992), 169–206.
  • L. Carleson and N. G. Makarov, “Some results connected with Brennan's conjecture”, Ark. Mat. 32:1 (1994), 33–62.
  • L. Carleson and V. Totik, “Hölder continuity of Green's functions”, Acta Sci. Math. $($Szeged$)$ 70:3–4 (2004), 557–608.
  • B.-Y. Chen, “Completeness of the Bergman metric on non-smooth pseudoconvex domains”, Ann. Polon. Math. 71:3 (1999), 241–251.
  • B.-Y. Chen, “Parameter dependence of the Bergman kernels”, Adv. Math. 299 (2016), 108–138.
  • B.-Y. Chen and S. Fu, “Comparison of the Bergman and Szegö kernels”, Adv. Math. 228:4 (2011), 2366–2384.
  • M. Christ, “On the $\bar\partial$ equation in weighted $L^2$ norms in $\C^1$”, J. Geom. Anal. 1:3 (1991), 193–230.
  • M. Christ, “Upper bounds for Bergman kernels associated to positive line bundles with smooth Hermitian metrics”, preprint, 2013.
  • H. Delin, “Pointwise estimates for the weighted Bergman projection kernel in $\C^n$, using a weighted $L^2$ estimate for the $\bar\partial$ equation”, Ann. Inst. Fourier $($Grenoble$)$ 48:4 (1998), 967–997.
  • J.-P. Demailly, “Mesures de Monge–Ampère et mesures pluriharmoniques”, Math. Z. 194:4 (1987), 519–564.
  • K. Diederich and J. E. Fornaess, “Pseudoconvex domains: bounded strictly plurisubharmonic exhaustion functions”, Invent. Math. 39:2 (1977), 129–141.
  • K. Diederich and J. E. Fornaess, “Proper holomorphic maps onto pseudoconvex domains with real-analytic boundary”, Ann. of Math. $(2)$ 110:3 (1979), 575–592.
  • K. Diederich and T. Ohsawa, “An estimate for the Bergman distance on pseudoconvex domains”, Ann. of Math. $(2)$ 141:1 (1995), 181–190.
  • H. Donnelly and C. Fefferman, “$L\sp{2}$-cohomology and index theorem for the Bergman metric”, Ann. of Math. $(2)$ 118:3 (1983), 593–618.
  • L. D. Edholm and J. D. McNeal, “The Bergman projection on fat Hartogs triangles: $L^p$ boundedness”, Proc. Amer. Math. Soc. 144:5 (2016), 2185–2196.
  • S. Fu and M.-C. Shaw, “The Diederich–Fornæss exponent and non-existence of Stein domains with Levi-flat boundaries”, J. Geom. Anal. 26:1 (2016), 220–230.
  • P. S. Harrington, “The order of plurisubharmonicity on pseudoconvex domains with Lipschitz boundaries”, Math. Res. Lett. 15:3 (2008), 485–490.
  • L. I. Hedberg, “Approximation in the mean by analytic functions”, Trans. Amer. Math. Soc. 163 (1972), 157–171.
  • G. Herbort, “The Bergman metric on hyperconvex domains”, Math. Z. 232:1 (1999), 183–196.
  • G. Herbort, “The pluricomplex Green function on pseudoconvex domains with a smooth boundary”, Internat. J. Math. 11:4 (2000), 509–522.
  • L. Hörmander, Notions of convexity, Progr. Math. 127, Birkhäuser, Boston, 1994.
  • M. Jarnicki and P. Pflug, Invariant distances and metrics in complex analysis, De Gruyter Expos. Math. 9, De Gruyter, Berlin, 1993.
  • S. G. Krantz and M. M. Peloso, “Analysis and geometry on worm domains”, J. Geom. Anal. 18:2 (2008), 478–510.
  • L. Lanzani, “Harmonic analysis techniques in several complex variables”, reprint, 2015.
  • L. Lempert, “On the boundary behavior of holomorphic mappings”, pp. 193–215 in Contributions to several complex variables, edited by A. Howard and P.-M. Wong, Asp. Math. E9, Vieweg, Braunschweig, 1986.
  • N. Lindholm, “Sampling in weighted $L^p$ spaces of entire functions in ${\mathbb C}^n$ and estimates of the Bergman kernel”, J. Funct. Anal. 182:2 (2001), 390–426.
  • X. Ma and G. Marinescu, Holomorphic Morse inequalities and Bergman kernels, Progr. Math. 254, Birkhäuser, Basel, 2007.
  • R. Mañé, P. Sad, and D. Sullivan, “On the dynamics of rational maps”, Ann. Sci. École Norm. Sup. $(4)$ 16:2 (1983), 193–217.
  • N. Nikolov and M. Trybuła, “The Kobayashi balls of ($\mathbb{C}$-)convex domains”, Monatsh. Math. 177:4 (2015), 627–635.
  • T. Ohsawa, “On the Bergman kernel of hyperconvex domains”, Nagoya Math. J. 129 (1993), 43–52.
  • T. Ohsawa and K. Takegoshi, “On the extension of $L^2$ holomorphic functions”, Math. Z. 195:2 (1987), 197–204.
  • E. A. Poletsky and M. I. Stessin, “Hardy and Bergman spaces on hyperconvex domains and their composition operators”, Indiana Univ. Math. J. 57:5 (2008), 2153–2201.
  • C. Pommerenke, “Uniformly perfect sets and the Poincaré metric”, Arch. Math. $($Basel$)$ 32:2 (1979), 192–199.
  • C. Pommerenke, Boundary behaviour of conformal maps, Grundlehren math. Wissenschaften 299, Springer, Berlin, 1992.
  • T. Ransford, Potential theory in the complex plane, London Math. Soc. Student Texts 28, Cambridge University, 1995.
  • M. Schiffer, “The kernel function of an orthonormal system”, Duke Math. J. 13 (1946), 529–540.
  • M. Skwarczyński, Biholomorphic invariants related to the Bergman function, Dissertationes Math. $($Rozprawy Mat.$)$ 173, Instytut Matematyczny Polskiej Akademii Nauk, Warsaw, 1980.
  • Z. Slodkowski, “Holomorphic motions and polynomial hulls”, Proc. Amer. Math. Soc. 111:2 (1991), 347–355.
  • V. Totik, Metric properties of harmonic measures, Mem. Amer. Math. Soc. 867, 2006.
  • J. B. Walsh, “Continuity of envelopes of plurisubharmonic functions”, J. Math. Mech. 18 (1968), 143–148.
  • S. Zelditch, “Off-diagonal decay of toric Bergman kernels”, Lett. Math. Phys. 106:12 (2016), 1849–1864.