Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The Bergman kernel function of
some Reinhardt domains

Authors: Sheng Gong and Xuean Zheng
Journal: Trans. Amer. Math. Soc. 348 (1996), 1771-1803
MSC (1991): Primary 32H10
MathSciNet review: 1329534
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The boundary behavior of the Bergman Kernel function of some Reinhardt domains is studied. Upper and lower bounds for the Bergman kernel function are found at the diagonal points $(z,\bar z)$. Let $D$ be the Reinhardt domain

\begin{displaymath}D=\left\{ z\in\mathbf C^n\,|\,\|z\|_\alpha=\sum_{j=1}^n|z_j|^{2/\al_j}<1\right\} \end{displaymath}

where $\al_j>0$, $j=1,2,\dots, n$; and let $K(z,\bar w)$ be the Bergman kernel function of $D$. Then there exist two positive constants $m$ and $M$ and a function $F$ such that

\begin{displaymath}mF(z,\bar z) \le K(z,\bar z)\le MF(z,\bar z) \end{displaymath}

holds for every $z\in D$. Here

\begin{displaymath}F(z,\bar z)=(-r(z))^{-n-1} \prod_{j=1}^n (-r(z)+|z_j|^{2/\al_j})^{1-\al_j} \end{displaymath}

and $r(z)=\|z\|_\alpha-1$ is the defining function for $D$. The constants $m$ and $M$ depend only on $\alpha=(\al_1,\dots, \al_n)$ and $n$, not on $z$.

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

  • 1. B. Bergman, Über die Kernfunktion eines Bereiches und ihr Verhalten am Rande, J. Reine Angew. Math. 169 (1933), 1--42.
  • 2. L. Boutet de Monvel and J. Sjöstrand, Sur la singularité des noyaux de Bergman et de Szegö, Astérisque 34--45, Soc. Math. de Paris, France, 1976, pp. 123--164. MR 58:28684
  • 3. D. Catlin, Estimates of invariant metrics on pseudoconvex domains of dimension two, Math. Z. 200 (1989), 429--466. MR 90e:32029
  • 4. J. P. D'Angelo, A note on the Bergman kernel, Duke Math. J. 45 (1978), 259--266. MR 57:12906
  • 5. ------, An explicit computation of the Bergman kernel function, J. Geom. Anal. 4 (1994), 23--34. MR 95a:32039
  • 6. C. Fefferman, The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent. Math. 26 (1974), 1--65. MR 50:2562
  • 7. J. McNeal, Boundary behavior of the Bergman kernel function in $\mathbf C^2$, Duke Math. J. 58 (1989), 499--512. MR 91c:32017
  • 8. ------, Local geometry of decoupled pseudoconvex domain, Aspekte der Math. E17 (1990), 223--230. MR 92g:32033
  • 9. A. Nagel, J. P. Rosay, E. M. Stein, and S. Wainger, Estimates for the Bergman and Szegö kernels in $\mathbf C^2$, Ann. of Math. 129 (1989), 113--149. MR 90g:32028

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 32H10

Retrieve articles in all journals with MSC (1991): 32H10

Additional Information

Sheng Gong
Affiliation: Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, People’s Republic of China; Department of Mathematics, University of California, San Diego, La Jolla, California 92093

Xuean Zheng
Affiliation: Department of Mathematics, Anhui University, Hefei, Anhui, 230039, People’s Republic of China

Received by editor(s): October 13, 1994
Article copyright: © Copyright 1996 American Mathematical Society