A counterexample to the differentiability of the Bergman kernel function
HTML articles powered by AMS MathViewer
- by So-Chin Chen PDF
- Proc. Amer. Math. Soc. 124 (1996), 1807-1810 Request permission
Abstract:
In this paper we prove the following main result. Let $D$ be a smoothly bounded pseudoconvex domain in $\mathbb {C}^n$ with $n\ge 2$. Suppose that there exists a complex variety sitting in the boundary $bD$; then we have \[ K_{D}(z,w)\notin C^{\infty }(\overline {D}\times \overline {D}-\Delta (bD)). \] 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
- David E. Barrett, Behavior of the Bergman projection on the Diederich-Fornæss worm, Acta Math. 168 (1992), no. 1-2, 1–10. MR 1149863, DOI 10.1007/BF02392975
- Steve Bell, Differentiability of the Bergman kernel and pseudolocal estimates, Math. Z. 192 (1986), no. 3, 467–472. MR 845219, DOI 10.1007/BF01164021
- Harold P. Boas, Extension of Kerzman’s theorem on differentiability of the Bergman kernel function, Indiana Univ. Math. J. 36 (1987), no. 3, 495–499. MR 905607, DOI 10.1512/iumj.1987.36.36027
- Harold P. Boas and Emil J. Straube, Sobolev estimates for the $\overline \partial$-Neumann operator on domains in $\textbf {C}^n$ admitting a defining function that is plurisubharmonic on the boundary, Math. Z. 206 (1991), no. 1, 81–88. MR 1086815, DOI 10.1007/BF02571327
- David 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. MR 740870, DOI 10.1090/pspum/041/740870
- So-Chin Chen, Global regularity of the $\overline \partial$-Neumann problem on circular domains, Math. Ann. 285 (1989), no. 1, 1–12. MR 1010187, DOI 10.1007/BF01442668
- So-Chin Chen, Global regularity of the $\overline \partial$-Neumann problem in dimension two, Several complex variables and complex geometry, Part 3 (Santa Cruz, CA, 1989) Proc. Sympos. Pure Math., vol. 52, Amer. Math. Soc., Providence, RI, 1991, pp. 55–61. MR 1128583
- John P. D’Angelo, Real hypersurfaces, orders of contact, and applications, Ann. of Math. (2) 115 (1982), no. 3, 615–637. MR 657241, DOI 10.2307/2007015
- Klas Diederich and John Erik Fornaess, Pseudoconvex domains: an example with nontrivial Nebenhülle, Math. Ann. 225 (1977), no. 3, 275–292. MR 430315, DOI 10.1007/BF01425243
- Norberto Kerzman, The Bergman kernel function. Differentiability at the boundary, Math. Ann. 195 (1972), 149–158. MR 294694, DOI 10.1007/BF01419622
- Peter Pflug, Quadratintegrable holomorphe Funktionen und die Serre-Vermutung, Math. Ann. 216 (1975), no. 3, 285–288 (German). MR 382717, DOI 10.1007/BF01430969
- Emil J. Straube, Exact regularity of Bergman, Szegő and Sobolev space projections in nonpseudoconvex domains, Math. Z. 192 (1986), no. 1, 117–128. MR 835396, DOI 10.1007/BF01162025
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
- Received by editor(s): December 1, 1994
- Communicated by: Eric Bedford
- © Copyright 1996 American Mathematical Society
- 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