대한수학회보 (Bulletin of the Korean Mathematical Society)

  • 제55권5호
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  • Pages.1483-1490
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  • 2018
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  • 1015-8634(pISSN)
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  • 2234-3016(eISSN)
대한수학회 (Korean Mathematical Society)
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ON THE LU QI-KENG PROBLEM FOR SLICE MONOGENIC FUNCTIONS

  • Xu, Zhenghua (School of Mathematics HeFei University of Technology)
  • 투고 : 2017.09.29
  • 심사 : 2018.03.08
  • 발행 : 2018.09.30
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⟨ 이전 논문 다음 논문 ⟩

초록

In this note, it is proven that the slice Bergman kernels for some axially symmetric slice domains are zero-free by a simple method.

키워드

참고문헌

  1. H. P. Boas, Counterexample to the Lu Qi-Keng conjecture, Proc. Amer. Math. Soc. 97 (1986), no. 2, 374-375. https://doi.org/10.1090/S0002-9939-1986-0835902-8
  2. F. Colombo, J. O. Gonzalez-Cervantes, and I. Sabadini, Further properties of the Bergman spaces of slice regular functions, Adv. Geom. 15 (2015), no. 4, 469-484.
  3. F. Colombo, I. Sabadini, and D. C. Struppa, Slice monogenic functions, Israel J. Math. 171 (2009), 385-403. https://doi.org/10.1007/s11856-009-0055-4
  4. F. Colombo, J. O. Gonzalez-Cervantes, M. E. Luna-Elizarraras, I. Sabadini, and M. Shapiro, On two approaches to the Bergman theory for slice regular functions, in Advances in hypercomplex analysis, 39-54, Springer INdAM Ser., 1, Springer, Milan, 2013.
  5. F. Colombo, J. O. Gonzalez-Cervantes, and I. Sabadini, The Bergman-Sce transform for slice monogenic functions, Math. Methods Appl. Sci. 34 (2011), no. 15, 1896-1909. https://doi.org/10.1002/mma.1489
  6. F. Colombo, J. O. Gonzalez-Cervantes, and I. Sabadini, On slice biregular functions and isomorphisms of Bergman spaces, Complex Var. Elliptic Equ. 57 (2012), no. 7-8, 825-839. https://doi.org/10.1080/17476933.2011.627441
  7. F. Colombo, J. O. Gonzalez-Cervantes, and I. Sabadini, The C-property for slice regular functions and applications to the Bergman space, Complex Var. Elliptic Equ. 58 (2013), no. 10, 1355-1372. https://doi.org/10.1080/17476933.2012.674521
  8. F. Colombo, I. Sabadini, and D. C. Struppa, Noncommutative Functional Calculus, Progress in Mathematics, 289, Birkhauser/Springer Basel AG, Basel, 2011.
  9. P. L. Duren, Univalent Functions, Grundlehren der Mathematischen Wissenschaften, 259, Springer-Verlag, New York, 1983.
  10. S. G. Gal and I. Sabadini, Approximation by polynomials on quaternionic compact sets, Math. Methods Appl. Sci. 38 (2015), no. 14, 3063-3074. https://doi.org/10.1002/mma.3281
  11. G. Gentili, C. Stoppato, and D. C. Struppa, Regular Functions of a Quaternionic Variable, Springer Monographs in Mathematics, Springer, Heidelberg, 2013.
  12. G. Gentili and D. C. Struppa, A new approach to Cullen-regular functions of a quaternionic variable, C. R. Math. Acad. Sci. Paris 342 (2006), no. 10, 741-744. https://doi.org/10.1016/j.crma.2006.03.015
  13. G. Gentili and D. C. Struppa, A new theory of regular functions of a quaternionic variable, Adv. Math. 216 (2007), no. 1, 279-301. https://doi.org/10.1016/j.aim.2007.05.010
  14. S. G. Krantz, Geometric Analysis of the Bergman Kernel and Metric, Graduate Texts in Mathematics, 268, Springer, New York, 2013.
  15. Q. Lu, On Kahler manifolds with constant curvature, Chinese Math. Acta 8 (1966), 283-298.
  16. J.-D. Park, On the zeros of the slice Bergman kernels for the upper half space and the unit ball on quaternions and Clifford algebras, Complex Anal. Oper. Theory 11 (2017), no. 2, 329-344. https://doi.org/10.1007/s11785-016-0550-7
  17. G. Ren, X. Wang, and Z. Xu, Slice regular functions on regular quadratic cones of real alternative algebras, in Modern trends in hypercomplex analysis, 227-245, Trends Math, Birkhauser/Springer, Cham, 2016.
  18. G. Ren and Z. Xu, Slice Lebesgue measure of quaternions, Adv. Appl. Clifford Algebr. 26 (2016), no. 1, 399-416. https://doi.org/10.1007/s00006-015-0578-1
  19. R. F. Rinehart, Elements of a theory of intrinsic functions on algebras, Duke Math. J 27 (1960), 1-19. https://doi.org/10.1215/S0012-7094-60-02701-0