Sobolev regularity of the $\overline{\partial }$ -equation on the Hartogs triangle

详细信息    查看全文

• 作者：Debraj Chakrabarti (1)
  Mei-Chi Shaw (2)
  
• 关键词：32W05 ; 32A07
• 刊名：Mathematische Annalen
• 出版年：2013
• 出版时间：May 2013
• 年：2013
• 卷：356
• 期：1
• 页码：241-258
• 全文大小：267KB
• 参考文献：1. Andreotti, A., Vesentini, E.: Carleman estimates for the Laplace鈥揃eltrami equation on complex manifolds. Inst. Hautes 脡tudes Sci. Publ. Math. 25, 81鈥?30 (1965)
  2. Barrett, D.: Behavior of the Bergman projection on the Diederich鈥揊ornaess worm. Acta Math. 168, 1鈥?0 (1992)
  3. Constantine, G.M., Savits, T.H.: A multivariate Fa脿 di Bruno formula with applications. Trans. Am. Math. Soc. 348(2), 503鈥?20 (1996) CrossRef
  4. Chakrabarti, D., Shaw, M.-C.: The Cauchy鈥揜iemann equations on product domains. Math. Ann. 349(4), 977鈥?98 (2011)
  5. Chaumat, J., Chollet, A.-M.: R茅gularit茅 h枚ld茅rienne de l鈥檕p茅rateur $\overline{\partial }$ -sur le triangle de Hartogs. Ann. Inst. Fourier (Grenoble) 41(4), 867鈥?82 (1991) CrossRef
  6. Chen, S.-C., Shaw, M.-C.: Partial differential equations in several complex variables. AMS/IP Studies in Advanced Mathematics, vol. 19. American Mathematical Society/International Press, Providence/Boston (2001)
  7. Dufresnoy, A.: Sur l鈥檕p茅rateur $d^{\prime \prime }$ et les fonctions diff茅rentiables au sens de Whitney. Ann. Inst. Fourier (Grenoble) 29, 229鈥?38 (1979) CrossRef
  8. Ehsani, D., Lieb, I.: $L^{p}$ -estimates for the Bergman projection on strictly pseudoconvex non-smooth domains. Math. Nachr. 281(7), 916鈥?29 (2008)
  9. Ehsani, D.: Integral representations on nonsmooth domains. Ill. J. Math. 53(4), 1127鈥?156 (2009)
  10. Ehsani, D.: Weighted $C^{k}$ -estimates for a class of integral operators on nonsmooth domains. Mich. Math. J. 59(3), 589鈥?20 (2010)
  11. H枚rmander, L.: $L^{2}$ estimates and existence theorems for the $\overline{\partial }$ -operator. Acta Math. 113, 89鈥?52 (1965)
  12. Horv谩th, J.: Topological vector spaces and distributions, vol. I. Addison-Wesley Publishing Co., Reading (1966)
  13. Kadison, R.V., Ringrose, J.R.: Fundamentals of the theory of operator algebras, vol. I. Graduate Studies in Mathematics, vol. 15. American Mathematical Society, Providence (1997)
  14. Kiselman, C.O.: A study of the Bergman projection in certain Hartogs domains. In: Proceedings in pure mathematics, vol. 52, part 3, pp. 219鈥?31 (1991)
  15. Kohn, J.J.: Global regularity for $\overline{\partial }$ on weakly pseudoconvex manifolds. Trans. Am. Math. Soc. 181, 273鈥?92 (1973)
  16. Ma, L., Michel, J.: $C^{k+\alpha }$ -estimates for the $\overline{\partial }$ -equation on the Hartogs triangle. Math. Ann. 294(4), 661鈥?75 (1992)
  17. Sibony, N.: Some aspects of weakly pseudoconvex domains. In: Proceedings of symposia in pure mathematics, vol. 52, Part I. American Mathematical Society, Providence, pp. 199鈥?31 (1991)
  18. Straube, E.J.: Lectures on the $\cal L^{2}$ -Sobolev theory of the $\overline{\partial }$ -Neumann problem. European Mathematical Society, Z眉rich (2010)
  
• 作者单位：Debraj Chakrabarti (1)
  Mei-Chi Shaw (2)
  
  1. TIFR Centre for Applicable Mathematics, Sharada Nagar, Chikkabommasandra, Bengaluru, 560065, India
  2. Department of Mathematics, University of Notre Dame, Notre Dame, IN, 46556, USA
  
• ISSN：1432-1807

文摘

The regularity of the $\overline{\partial }$ -problem on the domain $\{\left|{z_1}\right|\!<\!\left|{z_2}\right|\!<\!1\}$ in $\mathbb C ^2$ is studied using $L^2$ -methods. Estimates are obtained for the canonical solution in weighted $L^2$ -Sobolev spaces with a weight that is singular at the point $(0,0)$ . In particular, the singularity of the Bergman projection for the Hartogs triangle is contained at the singular point and it does not propagate.