Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A counterexample to the differentiability
of the Bergman kernel function


Author: So-Chin Chen
Journal: Proc. Amer. Math. Soc. 124 (1996), 1807-1810
MSC (1991): Primary 32H10
DOI: https://doi.org/10.1090/S0002-9939-96-03290-X
MathSciNet review: 1322916
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we prove the following main result. Let $D$ be a smoothly bounded pseudoconvex domain in $\mathbf {C} ^n$ with $n\ge 2$. Suppose that there exists a complex variety sitting in the boundary $bD$; then we have

\begin{displaymath}K_{D}(z,w)\notin C^{\infty }(\overline {D}\times \overline{D}-\Delta (bD)). \end{displaymath}

In particular, the Bergman kernel function associated with the Diederich-Fornaess worm domain is not smooth up to the boundary in joint variables off the diagonal of the boundary.


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

  • 1. D. Barrett, Behavior of the Bergman projection on the Diederich-Fornaess worm, Acta Math., 168, (1992), 1--10. MR 93c:32033
  • 2. S. Bell, Differentiability of the Bergman kernel and pseudo-local estimates, Math. Z., 192, (1986), 467--472. MR 87i:32034
  • 3. H. P. Boas, Extension of Kerzman's theorem on differentiability of the Bergman kernel function, Indiana Univ. Math. J., 36, (1987), 495--499. MR 88j:32028
  • 4. H. P. Boas and E. J. Straube, Sobolev estimates for the $\overline {\partial }$-Neumann operator on domains in $\mathbf {C} ^n$ admitting a defining function that is plurisubharmonic on the boundary, Math. Z., 206, (1991), 81--88. MR 92b:32027
  • 5. D. Catlin, Global regularity for the $\overline {\partial }$-Neumann problem, Proc. Sympos. Pure Math., vol. 41, Amer. Math. Soc., Providence, RI, 1984. MR 85j:32033
  • 6. S. C. Chen, Global regularity of the $\overline {\partial }$-Neumann problem on circular domains, Math. Ann., 285, (1989), 1--12. MR 90i:32028
  • 7. ------, Global regularity of the $\overline {\partial }$-Neumann problem in dimension two, Proc. Sympos. Pure Math., vol. 52, Part III, Amer. Math. Soc., Providence, RI, 1991, 55--61. MR 92h:32027
  • 8. J. D'Angelo, Real hypersurfaces, order of contact, and applications, Ann. of Math., 115, (1982), 615--637. MR 84a:32027
  • 9. K. Diederich and J. E. Fornaess, Pseudoconvex domains: an example with nontrivial Nebenhülle, Math. Ann., 225, (1977), 275--292. MR 55:3320
  • 10. N. Kerzman, The Bergman kernel function. Differentiability at the boundary, Math. Ann., 195, (1972), 149--158. MR 45:3762
  • 11. P. Pflug, Quadratintegrable holomorphe Funktionen und die Serre-Vermutung, Math. Ann., 216, (1975), 285--288. MR 52:3599
  • 12. E. J. Straube, Exact regularity of Bergman, Szegö and Sobolev space projections in non pseudoconvex domains, Math. Z., 192, (1986), 117--128. MR 87k:32045

Similar Articles

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

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


Additional Information

So-Chin Chen
Affiliation: Institute of Applied Mathematics, National Tsing Hua University, Hsinchu 30043, Taiwan, Republic of China
Email: scchen@am.nthu.edu.tw

DOI: https://doi.org/10.1090/S0002-9939-96-03290-X
Received by editor(s): December 1, 1994
Communicated by: Eric Bedford
Article copyright: © Copyright 1996 American Mathematical Society

American Mathematical Society