Duke Mathematical Journal

Boundary behavior of the Bergman kernel function in $\mathbb{C}^2$

Jeffery D. McNeal

Article information

Source
Duke Math. J. Volume 58, Number 2 (1989), 499-512.

Dates
First available in Project Euclid: 20 February 2004

http://projecteuclid.org/euclid.dmj/1077307535

Digital Object Identifier
doi:10.1215/S0012-7094-89-05822-5

Mathematical Reviews number (MathSciNet)
MR1016431

Zentralblatt MATH identifier
0675.32020

Subjects
Primary: 32H10
Secondary: 32F20

Citation

McNeal, Jeffery D. Boundary behavior of the Bergman kernel function in ℂ 2 . Duke Math. J. 58 (1989), no. 2, 499--512. doi:10.1215/S0012-7094-89-05822-5. http://projecteuclid.org/euclid.dmj/1077307535.

References

• [1] S. R. Bell, Biholomorphic mappings and the $\bar \partial$-problem, Ann. of Math. (2) 114 (1981), no. 1, 103–113.
• [2] S. R. Bell, Nonvanishing of the Bergman kernel function at boundary points of certain domains in ${\bf C}\sp{n}$, Math. Ann. 244 (1979), no. 1, 69–74.
• [3] S. Bergman, The kernel function and conformal mapping, American Mathematical Society, Providence, R.I., 1970.
• [4] T. Bloom and I. Graham, A geometric characterization of points of type $m$ on real submanifolds of ${\bf C}\sp{n}$, J. Differential Geometry 12 (1977), no. 2, 171–182.
• [5] D. Catlin, Necessary conditions for subellipticity of the $\bar \partial$-Neumann problem, Ann. of Math. (2) 117 (1983), no. 1, 147–171.
• [6] D. W. Catlin, Global regularity of the $\bar \partial$-Neumann problem, Complex analysis of several variables (Madison, Wis., 1982), Proc. Sympos. Pure Math., vol. 41, Amer. Math. Soc., Providence, RI, 1984, pp. 39–49.
• [7] D. Catlin, Estimates of invariant metrics on pseudoconvex domains of dimension two, Math. Z., to appear.
• [8] S.-C. Chen, Regularity of the Bergman projection on domains with partial transverse symmetries, Math. Ann. 277 (1987), no. 1, 135–140.
• [9] J. P. D'Angelo, Iterated commutators and derivatives of the Levi form, Complex analysis (University Park, Pa., 1986), Lecture Notes in Math., vol. 1268, Springer, Berlin, 1987, pp. 103–110.
• [10] K. Diederich, Das Randverhalten der Bergmanschen Kernfunktion und Metrik in streng pseudo-konvexen Gebieten, Math. Ann. 187 (1970), 9–36.
• [11] K. Diederich, Some recent developments in the theory of the Bergman kernel function: a survey, Several complex variables (Proc. Sympos. Pure Math., Vol. XXX, Part 1, Williams Coll., Williamstown, Mass., 1975), Amer. Math. Soc., Providence, R. I., 1977, pp. 127–137.
• [12] K. Diederich, J. E. Fornaess, and G. Herbort, Boundary behavior of the Bergman metric, Complex analysis of several variables (Madison, Wis., 1982), Proc. Sympos. Pure Math., vol. 41, Amer. Math. Soc., Providence, RI, 1984, pp. 59–67.
• [13] C. Fefferman, The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent. Math. 26 (1974), 1–65.
• [14] L. Hörmander, $L\sp{2}$ estimates and existence theorems for the $\bar \partial$ operator, Acta Math. 113 (1965), 89–152.
• [15] N. Kerzman, The Bergman kernel function. Differentiability at the boundary, Math. Ann. 195 (1972), 149–158.
• [16] J. J. Kohn, Boundary behavior of $\delta$ on weakly pseudo-convex manifolds of dimension two, J. Differential Geometry 6 (1972), 523–542.
• [17] J. J. Kohn, Pseudo-differential operators and non-elliptic problems, Pseudo-Diff. Operators (C.I.M.E., Stresa, 1968), Edizioni Cremonese, Rome, 1969, pp. 157–165.
• [18] J. J. Kohn and L. Nirenberg, Non-coercive boundary value problems, Comm. Pure Appl. Math. 18 (1965), 443–492.
• [19] A. Nagel, E. M. Stein, and S. Wainger, Balls and metrics defined by vector fields. I. Basic properties, Acta Math. 155 (1985), no. 1-2, 103–147.
• [20] A. Nagel, J. P. Rosay, E. M. Stein, and S. Wainger, Estimates for the Bergman and Szegö kernels in $\mathbb{C}^2$, preprint.
• [21] S. M. Webster, Biholomorphic mappings and the Bergman kernel off the diagonal, Invent. Math. 51 (1979), no. 2, 155–169.
• [22] C. L. Fefferman and J. J. Kohn, Hölder estimates on domains of complex dimension two and on three-dimensional CR manifolds, Adv. in Math. 69 (1988), no. 2, 223–303.
• [23] M. Christ, Regularity properties of the $\bar \partial_b$ equation on weakly pseudo-convex CR manifolds of dimension $3$, preprint.