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Mapping properties of weighted Bergman projection operators on Reinhardt domains

Authors: Željko Čučković and Yunus E. Zeytuncu
Journal: Proc. Amer. Math. Soc. 144 (2016), 3479-3491
MSC (2010): Primary 32A25; Secondary 32A26, 32A36
Published electronically: February 1, 2016
MathSciNet review: 3503715
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Abstract: We show that on smooth complete Reinhardt domains, weighted Bergman projection operators corresponding to exponentially decaying weights are unbounded on $ L^p$ spaces for all $ p\not =2$. On the other hand, we also show that the exponentially weighted projection operators are bounded on Sobolev spaces on the unit ball.

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  • [BG95] Aline Bonami and Sandrine Grellier, Weighted Bergman projections in domains of finite type in $ {\bf C}^2$, Harmonic analysis and operator theory (Caracas, 1994) Contemp. Math., vol. 189, Amer. Math. Soc., Providence, RI, 1995, pp. 65-80. MR 1347006 (96f:32042),
  • [Boa84] Harold P. Boas, Holomorphic reproducing kernels in Reinhardt domains, Pacific J. Math. 112 (1984), no. 2, 273-292. MR 743985 (85j:32038)
  • [BŞ12] David Barrett and Sönmez Şahutoğlu, Irregularity of the Bergman projection on worm domains in $ \mathbb{C}^n$, Michigan Math. J. 61 (2012), no. 1, 187-198. MR 2904008,
  • [CD06] Philippe Charpentier and Yves Dupain, Estimates for the Bergman and Szegö projections for pseudoconvex domains of finite type with locally diagonalizable Levi form, Publ. Mat. 50 (2006), no. 2, 413-446. MR 2273668 (2007h:32003),
  • [CDM13] P. Charpentier, Y. Dupain, and M. Mounkaila, Estimates for weighted Bergman projections on pseudo-convex domains of finite type in $ \Bbb {C}^n$,
    Preprint, arXiv:1212.1078v3, 2013.
  • [CDM14] P. Charpentier, Y. Dupain, and M. Mounkaila, On estimates for weighted Bergman projections, Preprint, arXiv:1403.3412v2, 2014.
  • [CL97] Der-Chen Chang and Bao Qin Li, Sobolev and Lipschitz estimates for weighted Bergman projections, Nagoya Math. J. 147 (1997), 147-178. MR 1475171 (98i:32035)
  • [Dos04] Milutin R. Dostanić, Unboundedness of the Bergman projections on $ L^p$ spaces with exponential weights, Proc. Edinb. Math. Soc. (2) 47 (2004), no. 1, 111-117. MR 2064739 (2005b:47067),
  • [Dos07] Milutin R. Dostanić, Integration operators on Bergman spaces with exponential weight, Rev. Mat. Iberoam. 23 (2007), no. 2, 421-436. MR 2371433 (2009b:47057),
  • [FR75] Frank Forelli and Walter Rudin, Projections on spaces of holomorphic functions in balls, Indiana Univ. Math. J. 24 (1974/75), 593-602. MR 0357866 (50 #10332)
  • [KP07] Steve G. Krantz and Marco M. Peloso, New results on the Bergman kernel of the worm domain in complex space, Electron. Res. Announc. Math. Sci. 14 (2007), 35-41 (electronic). MR 2336324 (2008d:32031)
  • [Kra01] Steven G. Krantz, Function theory of several complex variables, AMS Chelsea Publishing, Providence, RI, 2001. Reprint of the 1992 edition. MR 1846625 (2002e:32001)
  • [Lig89] Ewa Ligocka, On the Forelli-Rudin construction and weighted Bergman projections, Studia Math. 94 (1989), no. 3, 257-272. MR 1019793 (90i:32034)
  • [LS04] Loredana Lanzani and Elias M. Stein, Szegö and Bergman projections on non-smooth planar domains, J. Geom. Anal. 14 (2004), no. 1, 63-86. MR 2030575 (2004m:30063),
  • [MS94] J. D. McNeal and E. M. Stein, Mapping properties of the Bergman projection on convex domains of finite type, Duke Math. J. 73 (1994), no. 1, 177-199. MR 1257282 (94k:32037),
  • [PW90] Zbigniew Pasternak-Winiarski, On the dependence of the reproducing kernel on the weight of integration, J. Funct. Anal. 94 (1990), no. 1, 110-134. MR 1077547 (91j:46035),
  • [Str86] Emil J. Straube, Exact regularity of Bergman, Szegő and Sobolev space projections in nonpseudoconvex domains, Math. Z. 192 (1986), no. 1, 117-128. MR 835396 (87k:32045),
  • [Zey13a] Yunus E. Zeytuncu, $ L^p$ regularity of weighted Bergman projections, Trans. Amer. Math. Soc. 365 (2013), no. 6, 2959-2976. MR 3034455,
  • [Zey13b] Yunus E. Zeytuncu, Sobolev regularity of weighted Bergman projections on the unit disc, Complex Var. Elliptic Equ. 58 (2013), no. 3, 309-315. MR 3011926,

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Additional Information

Željko Čučković
Affiliation: Department of Mathematics and Statistics, University of Toledo, Toledo, Ohio 43606

Yunus E. Zeytuncu
Affiliation: Department of Mathematics and Statistics, University of Michigan, Dearborn, Dearborn, Michigan 48128

Keywords: Bergman projection, exponential weights, $L^p$ regularity, Sobolev regularity
Received by editor(s): September 3, 2015
Received by editor(s) in revised form: September 21, 2015, and October 11, 2015
Published electronically: February 1, 2016
Communicated by: Franc Forstneric
Article copyright: © Copyright 2016 American Mathematical Society

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