Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Failure of Korenblum's maximum principle in Bergman spaces with small exponents


Authors: Vladimir Božin and Boban Karapetrović
Journal: Proc. Amer. Math. Soc. 146 (2018), 2577-2584
MSC (2010): Primary 30H20
DOI: https://doi.org/10.1090/proc/13946
Published electronically: January 26, 2018
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The well-known conjecture due to B. Korenblum about the maximum principle in Bergman space $ A^p$ states that for $ 0<p<\infty $ there exists a constant $ 0<c<1$ with the following property. If $ f$ and $ g$ are holomorphic functions in the unit disk $ \mathbb{D}$ such that $ \vert f(z)\vert\leq \vert g(z)\vert$ for all $ c<\vert z\vert<1$, then $ \Vert f\Vert _{A^p}\leq \Vert g\Vert _{A^p}$. Hayman proved Korenblum's conjecture for $ p=2$, and Hinkkanen generalized this result by proving the conjecture for all $ 1\leq p<\infty $. The case $ 0<p<1$ of the conjecture has so far remained open. In this paper we resolve this remaining case of the conjecture by proving that Korenblum's maximum principle in Bergman space $ A^p$ does not hold when $ 0<p<1$.


References [Enhancements On Off] (What's this?)

  • [1] Peter Duren and Alexander Schuster, Bergman spaces, Mathematical Surveys and Monographs, vol. 100, American Mathematical Society, Providence, RI, 2004. MR 2033762
  • [2] Walter Kurt Hayman, On a conjecture of Korenblum, Analysis (Munich) 19 (1999), no. 2, 195-205. MR 1705360
  • [3] Haakan Hedenmalm, Boris Korenblum, and Kehe Zhu, Theory of Bergman spaces, Graduate Texts in Mathematics, vol. 199, Springer-Verlag, New York, 2000. MR 1758653
  • [4] A. Hinkkanen, On a maximum principle in Bergman space, J. Anal. Math. 79 (1999), 335-344. MR 1749317, https://doi.org/10.1007/BF02788246
  • [5] Boris Korenblum, A maximum principle for the Bergman space, Publ. Mat. 35 (1991), no. 2, 479-486. MR 1201570, https://doi.org/10.5565/PUBLMAT_35291_12
  • [6] B. Korenblum, R. O'Neil, K. Richards, and K. Zhu, Totally monotone functions with applications to the Bergman space, Trans. Amer. Math. Soc. 337 (1993), no. 2, 795-806. MR 1118827, https://doi.org/10.2307/2154243
  • [7] Boris Korenblum and Kendall Richards, Majorization and domination in the Bergman space, Proc. Amer. Math. Soc. 117 (1993), no. 1, 153-158. MR 1113643, https://doi.org/10.2307/2159710
  • [8] Jerk Matero, On Korenblum's maximum principle for the Bergman space, Arch. Math. (Basel) 64 (1995), no. 4, 337-340. MR 1319004, https://doi.org/10.1007/BF01198089
  • [9] Alexander Schuster, The maximum principle for the Bergman space and the Möbius pseudodistance for the annulus, Proc. Amer. Math. Soc. 134 (2006), no. 12, 3525-3530. MR 2240664, https://doi.org/10.1090/S0002-9939-06-08378-X
  • [10] Wilhelm Schwick, On Korenblum's maximum principle, Proc. Amer. Math. Soc. 125 (1997), no. 9, 2581-2587. MR 1307563, https://doi.org/10.1090/S0002-9939-97-03247-4
  • [11] Chunjie Wang, Refining the constant in a maximum principle for the Bergman space, Proc. Amer. Math. Soc. 132 (2004), no. 3, 853-855. MR 2019965, https://doi.org/10.1090/S0002-9939-03-07137-5
  • [12] Chunjie Wang, On Korenblum's maximum principle, Proc. Amer. Math. Soc. 134 (2006), no. 7, 2061-2066. MR 2215775, https://doi.org/10.1090/S0002-9939-06-08311-0
  • [13] Chunjie Wang, On a maximum principle for Bergman spaces with small exponents, Integral Equations Operator Theory 59 (2007), no. 4, 597-601. MR 2370052, https://doi.org/10.1007/s00020-007-1539-4
  • [14] Chunjie Wang, Domination in the Bergman space and Korenblum's constant, Integral Equations Operator Theory 61 (2008), no. 3, 423-432. MR 2417506, https://doi.org/10.1007/s00020-008-1587-4
  • [15] Chunjie Wang, Some results on Korenblum's maximum principle, J. Math. Anal. Appl. 373 (2011), no. 2, 393-398. MR 2720689, https://doi.org/10.1016/j.jmaa.2010.07.052

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 30H20

Retrieve articles in all journals with MSC (2010): 30H20


Additional Information

Vladimir Božin
Affiliation: Faculty of Mathematics University of Belgrade Studentski trg 16 Serbia
Email: bozinv@mi.sanu.ac.rs

Boban Karapetrović
Affiliation: Faculty of Mathematics University of Belgrade Studentski trg 16 Serbia
Email: bkarapetrovic@matf.bg.ac.rs

DOI: https://doi.org/10.1090/proc/13946
Keywords: Bergman spaces, Korenblum's maximum principle
Received by editor(s): May 31, 2017
Received by editor(s) in revised form: September 3, 2017
Published electronically: January 26, 2018
Additional Notes: The authors were supported by NTR Serbia, Project ON174032
Communicated by: Stephan Ramm Garcia
Article copyright: © Copyright 2018 American Mathematical Society

American Mathematical Society