Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

Lp SOBOLEV MAPPING PROPERTIES OF THE BERGMAN PROJECTIONS ON n-DIMENSIONAL GENERALIZED HARTOGS TRIANGLES

  • Zhang, Shuo (College of Science Tianjin University of Technology)
  • Received : 2020.09.07
  • Accepted : 2021.08.25
  • Published : 2021.11.30
Download PDF
⟨ Previous Next ⟩

Abstract

The n-dimensional generalized Hartogs triangles ℍn𝐩 with n ≥ 2 and 𝐩 := (p1, …, pn) ∈ (ℝ+)n are the domains defined by ℍn𝐩 := {z = (z1, …, zn) ∈ ℂn : |z1|p1 < ⋯ < |zn|pn < 1}. In this paper, we study the Lp Sobolev mapping properties for the Bergman projections on the n-dimensional generalized Hartogs triangles ℍn𝐩, which can be viewed as a continuation of the work by S. Zhang in [25] and a higher-dimensional generalization of the work by L. D. Edholm and J. D. McNeal in [16].

Keywords

Acknowledgement

The author thanks his Ph.D. advisor Prof. Feng Rong for helpful comments and suggestions to this manuscript. The author is partially supported by the National Natural Science Foundation of China (Grant No. 11871333).

References

  1. D. E. Barrett, Behavior of the Bergman projection on the Diederich-Fornaess worm, Acta Math. 168 (1992), no. 1-2, 1-10. https://doi.org/10.1007/BF02392975
  2. T. Beberok, Lp boundedness of the Bergman projection on some generalized Hartogs triangles, Bull. Iranian Math. Soc. 43 (2017), no. 7, 2275-2280.
  3. S. R. Bell, Biholomorphic mappings and the $\bar{\partial}$-problem, Ann. of Math. (2) 114 (1981), no. 1, 103-113. https://doi.org/10.2307/1971379
  4. S. Bell and E. Ligocka, A simplification and extension of Fefferman's theorem on biholomorphic mappings, Invent. Math. 57 (1980), no. 3, 283-289. https://doi.org/10.1007/BF01418930
  5. H. P. Boas and E. J. Straube, Sobolev estimates for the $\bar{\partial}$-Neumann operator on domains in Cn admitting a defining function that is plurisubharmonic on the boundary, Math. Z. 206 (1991), no. 1, 81-88. https://doi.org/10.1007/BF02571327
  6. H. P. Boas and E. J. Straube, Global regularity of the $\bar{\partial}$-Neumann problem: a survey of the L2-Sobolev theory, in Several complex variables (Berkeley, CA, 1995-1996), 79-111, Math. Sci. Res. Inst. Publ., 37, Cambridge Univ. Press, Cambridge, 1999.
  7. P. Charpentier and Y. Dupain, Estimates for the Bergman and Szego projections for pseudoconvex domains of finite type with locally diagonalizable Levi form, Publ. Mat. 50 (2006), no. 2, 413-446. https://doi.org/10.5565/PUBLMAT_50206_08
  8. P. Charpentier, Y. Dupain, and M. Mounkaila, On estimates for weighted Bergman projections, Proc. Amer. Math. Soc. 143 (2015), no. 12, 5337-5352. https://doi.org/10.1090/proc/12660
  9. D. Chakrabarti, L. D. Edholm, and J. D. McNeal, Duality and approximation of Bergman spaces, Adv. Math. 341 (2019), 616-656. https://doi.org/10.1016/j.aim.2018.10.041
  10. D. Chakrabarti and Y. E. Zeytuncu, Lp mapping properties of the Bergman projection on the Hartogs triangle, Proc. Amer. Math. Soc. 144 (2016), no. 4, 1643-1653. https://doi.org/10.1090/proc/12820
  11. L. Chen, The Lp boundedness of the Bergman projection for a class of bounded Hartogs domains, J. Math. Anal. Appl. 448 (2017), no. 1, 598-610. https://doi.org/10.1016/j.jmaa.2016.11.024
  12. L. Chen, Weighted Sobolev regularity of the Bergman projection on the Hartogs triangle, Pacific J. Math. 288 (2017), no. 2, 307-318. https://doi.org/10.2140/pjm.2017.288.307
  13. L. D. Edholm, Bergman theory of certain generalized Hartogs triangles, Pacific J. Math. 284 (2016), no. 2, 327-342. https://doi.org/10.2140/pjm.2016.284.327
  14. L. D. Edholm and J. D. McNeal, The Bergman projection on fat Hartogs triangles: Lp boundedness, Proc. Amer. Math. Soc. 144 (2016), no. 5, 2185-2196. https://doi.org/10.1090/proc/12878
  15. L. D. Edholm and J. D. McNeal, Bergman subspaces and subkernels: degenerate Lp mapping and zeroes, J. Geom. Anal. 27 (2017), no. 4, 2658-2683. https://doi.org/10.1007/s12220-017-9777-4
  16. L. D. Edholm and J. D. McNeal, Sobolev mapping of some holomorphic projections, J. Geom. Anal. 30 (2020), no. 2, 1293-1311. https://doi.org/10.1007/s12220-019-00345-6
  17. S. G. Krantz, Geometric Analysis of the Bergman Kernel and Metric, Graduate Texts in Mathematics, 268, Springer, New York, 2013. https://doi.org/10.1007/978-1-4614-7924-6
  18. J. D. McNeal, Boundary behavior of the Bergman kernel function in C2, Duke Math. J. 58 (1989), no. 2, 499-512. https://doi.org/10.1215/S0012-7094-89-05822-5
  19. J. D. McNeal, Local geometry of decoupled pseudoconvex domains, in Complex analysis (Wuppertal, 1991), 223-230, Aspects Math., E17, Friedr. Vieweg, Braunschweig, 1991.
  20. J. D. McNeal, Estimates on the Bergman kernels of convex domains, Adv. Math. 109 (1994), no. 1, 108-139. https://doi.org/10.1006/aima.1994.1082
  21. A. Nagel, J. Rosay, E. M. Stein, and S. Wainger, Estimates for the Bergman and Szegokernels in C2, Ann. of Math. (2) 129 (1989), no. 1, 113-149. https://doi.org/10.2307/1971487
  22. J.-D. Park, The explicit forms and zeros of the Bergman kernel for 3-dimensional Hartogs triangles, J. Math. Anal. Appl. 460 (2018), no. 2, 954-975. https://doi.org/10.1016/j.jmaa.2017.12.002
  23. D. H. Phong and E. M. Stein, Estimates for the Bergman and Szego projections on strongly pseudo-convex domains, Duke Math. J. 44 (1977), no. 3, 695-704. http://projecteuclid.org/euclid.dmj/1077312391 https://doi.org/10.1215/S0012-7094-77-04429-5
  24. Y. Tang and Z. Tu, Special Toeplitz operators on a class of bounded Hartogs domains, Arch. Math. (Basel) 114 (2020), no. 6, 661-675. https://doi.org/10.1007/s00013-019-01424-4
  25. S. Zhang, Lp boundedness for the Bergman projections over n-dimensional generalized Hartogs triangles, Complex Var. Elliptic Equ., 2020. https://doi.org/10.1080/17476933.2020.1769085
  26. S. Zhang, Mapping properties of the Bergman projections on elementary Reinhardt domains, Math. Slovaca 71 (2021), no. 4, 831-844. https://doi.org/10.1515/ms-2021-0024