The hyper-singular cousin of the Bergman projection
HTML articles powered by AMS MathViewer
- by Guozheng Cheng, Xiang Fang, Zipeng Wang and Jiayang Yu PDF
- Trans. Amer. Math. Soc. 369 (2017), 8643-8662 Request permission
Abstract:
The Bergman projection over the unit disk is one of the most studied objects in complex analysis and operator theory. In this paper the new finding is to observe some unexpected patterns in boundedness when we consider the hyper-singular cousins of the Bergman projection. We also show that these patterns don’t hold for the half-space, but we conjecture that they hold for general bounded domains.References
- N. Arcozzi, R. Rochberg, and E. Sawyer, Carleson measures for the Drury-Arveson Hardy space and other Besov-Sobolev spaces on complex balls, Adv. Math. 218 (2008), no. 4, 1107–1180. MR 2419381, DOI 10.1016/j.aim.2008.03.001
- Stephen M. Buckley, Pekka Koskela, and Dragan Vukotić, Fractional integration, differentiation, and weighted Bergman spaces, Math. Proc. Cambridge Philos. Soc. 126 (1999), no. 2, 369–385. MR 1670257, DOI 10.1017/S030500419800334X
- Aline Bonami, Gustavo Garrigós, and Cyrille Nana, $L^p$-$L^q$ estimates for Bergman projections in bounded symmetric domains of tube type, J. Geom. Anal. 24 (2014), no. 4, 1737–1769. MR 3261716, DOI 10.1007/s12220-013-9393-x
- G. Cheng, X. Fang, Z. Wang, and J. Yu, Three measure-theoretic problems for the complex Riesz potential operator, preprint, 2014.
- G. Cheng, X. Fang, Z. Wang, and J. Yu, Endpoint estimates for the complex Riesz potential operator, preprint, 2015.
- D. Cruz-Uribe, Two weight norm inequalities for fractional integral operators and commutators, preprint, 2014.
- Yaohua Deng, Li Huang, Tao Zhao, and Dechao Zheng, Bergman projection and Bergman spaces, J. Operator Theory 46 (2001), no. 1, 3–24. MR 1862176
- Omar El-Fallah, Karim Kellay, Javad Mashreghi, and Thomas Ransford, A primer on the Dirichlet space, Cambridge Tracts in Mathematics, vol. 203, Cambridge University Press, Cambridge, 2014. MR 3185375
- Peter Duren and Alexander Schuster, Bergman spaces, Mathematical Surveys and Monographs, vol. 100, American Mathematical Society, Providence, RI, 2004. MR 2033762, DOI 10.1090/surv/100
- Loukas Grafakos, Classical Fourier analysis, 2nd ed., Graduate Texts in Mathematics, vol. 249, Springer, New York, 2008. MR 2445437
- José García-Cuerva and A. Eduardo Gatto, Boundedness properties of fractional integral operators associated to non-doubling measures, Studia Math. 162 (2004), no. 3, 245–261. MR 2047654, DOI 10.4064/sm162-3-5
- Charles Fefferman, Inequalities for strongly singular convolution operators, Acta Math. 124 (1970), 9–36. MR 257819, DOI 10.1007/BF02394567
- H. Turgay Kaptanoğlu, Carleson measures for Besov spaces on the ball with applications, J. Funct. Anal. 250 (2007), no. 2, 483–520. MR 2352489, DOI 10.1016/j.jfa.2006.12.016
- Walter Rudin, Function theory in the unit ball of $\textbf {C}^{n}$, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 241, Springer-Verlag, New York-Berlin, 1980. MR 601594
- T. Tao, Harmonic analysis, Lecture notes at UCLA, http://www.math.ucla.edu/$\thicksim$tao/ 247a.1.06f/notes2.pdf.
- P. Wojtaszczyk, On multipliers into Bergman spaces and Nevanlinna class, Canad. Math. Bull. 33 (1990), no. 2, 151–161. MR 1060368, DOI 10.4153/CMB-1990-026-7
- Ruhan Zhao, Generalization of Schur’s test and its application to a class of integral operators on the unit ball of $\Bbb {C}^n$, Integral Equations Operator Theory 82 (2015), no. 4, 519–532. MR 3369311, DOI 10.1007/s00020-014-2215-0
- Ke He Zhu, Bloch type spaces of analytic functions, Rocky Mountain J. Math. 23 (1993), no. 3, 1143–1177. MR 1245472, DOI 10.1216/rmjm/1181072549
- Kehe Zhu, Spaces of holomorphic functions in the unit ball, Graduate Texts in Mathematics, vol. 226, Springer-Verlag, New York, 2005. MR 2115155
- Kehe Zhu, Operator theory in function spaces, 2nd ed., Mathematical Surveys and Monographs, vol. 138, American Mathematical Society, Providence, RI, 2007. MR 2311536, DOI 10.1090/surv/138
Additional Information
- Guozheng Cheng
- Affiliation: School of Mathematics, Wenzhou University, Wenzhou 325035, People’s Republic of China
- MR Author ID: 795828
- Email: gzhcheng@wzu.edu.cn
- Xiang Fang
- Affiliation: Department of Mathematics, National Central University, Chung-Li 32001, Taiwan
- MR Author ID: 711208
- Email: xfang@math.ncu.edu.tw
- Zipeng Wang
- Affiliation: College of Mathematics and Information Sciences, Shaanxi Normal University, Xi’an 710062, People’s Republic of China – and – Department of Mathematics, National Central University, Chung-Li 32001, Taiwan
- Email: zipengwang11@fudan.edu.cn
- Jiayang Yu
- Affiliation: School of Mathematics, Sichuan University, Chengdu 610064, People’s Republic of China
- MR Author ID: 1050786
- Email: jiayangyu@scu.edu.cn
- Received by editor(s): June 9, 2015
- Received by editor(s) in revised form: December 9, 2015, and February 12, 2016
- Published electronically: May 30, 2017
- Additional Notes: The first author was supported by NSFC (11471249) and Zhejiang Provincial NSFC (LY14A010021)
The second author was supported by NSC of Taiwan (102-2115-M-008-016-MY2)
The third author was supported by NSFC (11371096).
The fourth author was supported by NSFC (11271075). - © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 369 (2017), 8643-8662
- MSC (2010): Primary 47B34, 47G10
- DOI: https://doi.org/10.1090/tran/6923
- MathSciNet review: 3710638