대한수학회지 (Journal of the Korean Mathematical Society)

  • 제58권6호
  • /
  • Pages.1347-1365
  • /
  • 2021
  • /
  • 0304-9914(pISSN)
  • /
  • 2234-3008(eISSN)
대한수학회 (Korean Mathematical Society)
권호 표지
DOI QR코드

DOI QR Code

BERGMAN TYPE OPERATORS ON SOME GENERALIZED CARTAN-HARTOGS DOMAINS

  • He, Le (School of Mathematics and Statistics Wuhan University) ;
  • Tang, Yanyan (School of Mathematics and Statistics Henan University) ;
  • Tu, Zhenhan (School of Mathematics and Statistics Wuhan University)
  • 투고 : 2020.12.23
  • 심사 : 2021.08.20
  • 발행 : 2021.11.01
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

For µ = (µ1, …, µt) (µj > 0), ξ = (z1, …, zt, w) ∈ ℂn1 × … × ℂnt × ℂm, define $${\Omega}({\mu},t)=\{{\xi}{\in}\mathbb{B}_{n_1}{\times}{\cdots}{\times}\mathbb{B}_{n_t}{\times}\mathbb{C}^m:{\parallel}w{\parallel}^2 where $\mathbb{B}_{n_j}$ is the unit ball in ℂnj (1 ≤ j ≤ t), C(χ, µ) is a constant only depending on χ = (n1, …, nt) and µ = (µ1, …, µt), which is a special type of generalized Cartan-Hartogs domain. We will give some sufficient and necessary conditions for the boundedness of some type of operators on Lp(Ω(µ, t), ω) (the weighted Lp space of Ω(µ, t) with weight ω, 1 < p < ∞). This result generalizes the works from certain classes of generalized complex ellipsoids to the generalized Cartan-Hartogs domain Ω(µ, t).

키워드

과제정보

This work was supported by the National Natural Science Foundation of China (No.12071354).

참고문헌

  1. H. Ahn and J.-D. Park, The explicit forms and zeros of the Bergman kernel function for Hartogs type domains, J. Funct. Anal. 262 (2012), no. 8, 3518-3547. https://doi.org/10.1016/j.jfa.2012.01.021
  2. E. Bi, Z. Feng, and Z. Tu, Balanced metrics on the Fock-Bargmann-Hartogs domains, Ann. Global Anal. Geom. 49 (2016), no. 4, 349-359. https://doi.org/10.1007/s10455-016-9495-3
  3. E. Bi and Z. Tu, Rigidity of proper holomorphic mappings between generalized Fock-Bargmann-Hartogs domains, Pacific J. Math. 297 (2018), no. 2, 277-297. https://doi.org/10.2140/pjm.2018.297.277
  4. P. Charpentier, Y. Dupain, and M. Mounkaila, Estimates for weighted Bergman projections on pseudo-convex domains of finite type in ℂn, Complex Var. Elliptic Equ. 59 (2014), no. 8, 1070-1095. https://doi.org/10.1080/17476933.2013.805411
  5. J. P. D'Angelo, An explicit computation of the Bergman kernel function, J. Geom. Anal. 4 (1994), no. 1, 23-34. https://doi.org/10.1007/BF02921591
  6. F. Forelli and W. Rudin, Projections on spaces of holomorphic functions in balls, Indiana Univ. Math. J. 24 (1974/75), 593-602. https://doi.org/10.1512/iumj.1974.24.24044
  7. S. G. Gindikin, Analysis in homogeneous domains, Russ. Math. Surv. 19(1964), 1-89. https://doi.org/10.1070/RM1964v019n04ABEH001153
  8. H. Ishi, J.-D. Park, and A. Yamamori, Bergman kernel function for Hartogs domains over bounded homogeneous domains, J. Geom. Anal. 27 (2017), no. 2, 1703-1736. https://doi.org/10.1007/s12220-016-9737-4
  9. H. T. Kaptanoglu, Bergman projections on Besov spaces on balls, Illinois J. Math. 49 (2005), no. 2, 385-403. http://projecteuclid.org/euclid.ijm/1258138024 https://doi.org/10.1215/ijm/1258138024
  10. E. Ligocka, On the Forelli-Rudin construction and weighted Bergman projections, Studia Math. 94 (1989), no. 3, 257-272. https://doi.org/10.4064/sm-94-3-257-272
  11. S. H. Liu and M. Stoll, Projections on spaces of holomorphic functions on certain domains in ℂ2, Complex Variables Theory Appl. 17 (1992), no. 3-4, 223-233. https://doi.org/10.1080/17476939208814515
  12. N. Nikolov and W. Zwonek, The Bergman kernel of the symmetrized polydisc in higher dimensions has zeros, Arch. Math. (Basel) 87 (2006), no. 5, 412-416. https://doi.org/10.1007/s00013-006-1801-z
  13. K. Oeljeklaus, P. Pflug, and E. H. Youssfi, The Bergman kernel of the minimal ball and applications, Ann. Inst. Fourier (Grenoble) 47 (1997), no. 3, 915-928. https://doi.org/10.5802/aif.1585
  14. J.-D. Park, New formulas of the Bergman kernels for complex ellipsoids in ℂ2, Proc. Amer. Math. Soc. 136 (2008), no. 12, 4211-4221. https://doi.org/10.1090/S0002-9939-08-09576-2
  15. J.-D. Park, Explicit formulas of the Bergman kernel for 3-dimensional complex ellipsoids, J. Math. Anal. Appl. 400 (2013), no. 2, 664-674. https://doi.org/10.1016/j.jmaa.2012.11.017
  16. Z. Pasternak-Winiarski, On the dependence of the reproducing kernel on the weight of integration, J. Funct. Anal. 94 (1990), no. 1, 110-134. https://doi.org/10.1016/0022-1236(90)90030-O
  17. J. Shi, Bergman type operators on a class of weakly pseudoconvex domains, Sci. China Ser. A 41 (1998), no. 1, 22-32. https://doi.org/10.1007/BF02900768
  18. A. Yamamori, The Bergman kernel of the Fock-Bargmann-Hartogs domain and the polylogarithm function, Complex Var. Elliptic Equ. 58 (2013), no. 6, 783-793. https://doi.org/10.1080/17476933.2011.620098
  19. W. Yin, The Bergman kernel on four tyoes of suoer-Cartan domains, Chinese Sci. Bull. 44 (1999), 1391-1395. https://doi.org/10.1360/csb1999-44-13-1391
  20. W. Yin, The Bergman kernels on super-Cartan domains of the first type, Sci. China Ser. A 43 (2000), no. 1, 13-21. https://doi.org/10.1007/BF02903843
  21. W. Yin, K. Lu, and G. Roos, New classes of domains with explicit Bergman kernel, Sci. China Ser. A 47 (2004), no. 3, 352-371. https://doi.org/10.1360/03ys0090
  22. E. H. Youssfi, Proper holomorphic liftings and new formulas for the Bergman and Szego kernels, Studia Math. 152 (2002), no. 2, 161-186. https://doi.org/10.4064/sm152-2-5
  23. R. Zhao, Generalization of Schur's test and its application to a class of integral operators on the unit ball of ℂn, Integral Equations Operator Theory 82 (2015), no. 4, 519-532. https://doi.org/10.1007/s00020-014-2215-0
  24. K. Zhu, The Bergman spaces, the Bloch space, and Gleason's problem, Trans. Amer. Math. Soc. 309 (1988), no. 1, 253-268. https://doi.org/10.2307/2001168
  25. K. Zhu, A Forelli-Rudin type theorem with applications, Complex Variables Theory Appl. 16 (1991), no. 2-3, 107-113. https://doi.org/10.1080/17476939108814472
  26. K. Zhu, Operator theory in function spaces, second edition, Mathematical Surveys and Monographs, 138, American Mathematical Society, Providence, RI, 2007. https://doi.org/10.1090/surv/138