Mapping properties of operator-valued Bergman projections
HTML articles powered by AMS MathViewer
- by Liang Wang, Bang Xu and Dejian Zhou;
- Proc. Amer. Math. Soc. 151 (2023), 1221-1234
- DOI: https://doi.org/10.1090/proc/16213
- Published electronically: December 21, 2022
- HTML | PDF | Request permission
Abstract:
In this paper, we study the boundedness theory for Bergman projection in the operator-valued setting. More precisely, let $\mathbb {D}$ be the open unit disk in the complex plane $\mathbb {C}$ and $\mathcal {M}$ be a semifinite von Neumann algebra. We prove that \begin{equation*} \|P(f)\|_{L_{1,\infty }(\mathcal {N})}\leq C \|f\|_{L_1(\mathcal {N})}, \end{equation*} where $\mathcal {N}=L_{\infty }(\mathbb {D})\bar {\otimes }\mathcal {M}$ and $P$ denotes the Bergman projection. Consequently, $P$ is bounded on $L_{p}(\mathcal {N})$ with $1<p<\infty$. As applications, we also obtain Kolmogorov and Zygmund inequalities for the Bergman projection $P$.References
- Sheldon Axler, Bergman spaces and their operators, Surveys of some recent results in operator theory, Vol. I, Pitman Res. Notes Math. Ser., vol. 171, Longman Sci. Tech., Harlow, 1988, pp. 1–50. MR 958569
- Stefan Bergman, The Kernel Function and Conformal Mapping, Mathematical Surveys, No. 5, American Mathematical Society, New York, 1950. MR 38439, DOI 10.1090/surv/005
- L. Cadilhac, personal communication, 2019.
- Léonard Cadilhac, José M. Conde-Alonso, and Javier Parcet, Spectral multipliers in group algebras and noncommutative Calderón-Zygmund theory, J. Math. Pures Appl. (9) 163 (2022), 450–472 (English, with English and French summaries). MR 4438906, DOI 10.1016/j.matpur.2022.05.011
- M. Caspers, D. Potapov, F. Sukochev, and D. Zanin, Weak type commutator and Lipschitz estimates: resolution of the Nazarov-Peller conjecture, Amer. J. Math. 141 (2019), no. 3, 593–610. MR 3956516, DOI 10.1353/ajm.2019.0019
- Zeqian Chen, Quanhua Xu, and Zhi Yin, Harmonic analysis on quantum tori, Comm. Math. Phys. 322 (2013), no. 3, 755–805. MR 3079331, DOI 10.1007/s00220-013-1745-7
- I. Cuculescu, Martingales on von Neumann algebras, J. Multivariate Anal. 1 (1971), no. 1, 17–27. MR 295398, DOI 10.1016/0047-259X(71)90027-3
- 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
- Peter G. Dodds, Theresa K.-Y. Dodds, and Ben de Pagter, Noncommutative Köthe duality, Trans. Amer. Math. Soc. 339 (1993), no. 2, 717–750. MR 1113694, DOI 10.1090/S0002-9947-1993-1113694-3
- Thierry Fack and Hideki Kosaki, Generalized $s$-numbers of $\tau$-measurable operators, Pacific J. Math. 123 (1986), no. 2, 269–300. MR 840845, DOI 10.2140/pjm.1986.123.269
- Frank Forelli and Walter Rudin, Projections on spaces of holomorphic functions in balls, Indiana Univ. Math. J. 24 (1974/75), 593–602. MR 357866, DOI 10.1512/iumj.1974.24.24044
- G. Hong, X. Lai, and B. Xu, Maximal singular integral operators acting on noncommutative $L_p$-spaces, Math. Ann. (2022), DOI 10.1007/s00208-022-02401-z.
- Guixiang Hong, Honghai Liu, and Tao Mei, An operator-valued $T1$ theory for symmetric CZOs, J. Funct. Anal. 278 (2020), no. 7, 108420, 27. MR 4053624, DOI 10.1016/j.jfa.2019.108420
- Guixiang Hong, Luis Daniel López-Sánchez, José María Martell, and Javier Parcet, Calderón-Zygmund operators associated to matrix-valued kernels, Int. Math. Res. Not. IMRN 5 (2014), 1221–1252. MR 3178596, DOI 10.1093/imrn/rns250
- Ying Hu, Noncommutative extrapolation theorems and applications, Illinois J. Math. 53 (2009), no. 2, 463–482. MR 2594639
- Yong Jiao, Narcisse Randrianantoanina, Lian Wu, and Dejian Zhou, Square functions for noncommutative differentially subordinate martingales, Comm. Math. Phys. 374 (2020), no. 2, 975–1019. MR 4072235, DOI 10.1007/s00220-019-03391-x
- Marius Junge, Tao Mei, and Javier Parcet, Smooth Fourier multipliers on group von Neumann algebras, Geom. Funct. Anal. 24 (2014), no. 6, 1913–1980. MR 3283931, DOI 10.1007/s00039-014-0307-2
- Jeffery D. McNeal, The Bergman projection as a singular integral operator, J. Geom. Anal. 4 (1994), no. 1, 91–103. MR 1274139, DOI 10.1007/BF02921594
- N. J. Kalton and F. A. Sukochev, Symmetric norms and spaces of operators, J. Reine Angew. Math. 621 (2008), 81–121. MR 2431251, DOI 10.1515/CRELLE.2008.059
- Steven Lord, Fedor Sukochev, and Dmitriy Zanin, Singular traces, De Gruyter Studies in Mathematics, vol. 46, De Gruyter, Berlin, 2013. Theory and applications. MR 3099777
- Tao Mei, Operator valued Hardy spaces, Mem. Amer. Math. Soc. 188 (2007), no. 881, vi+64. MR 2327840, DOI 10.1090/memo/0881
- Tao Mei and Javier Parcet, Pseudo-localization of singular integrals and noncommutative Littlewood-Paley inequalities, Int. Math. Res. Not. IMRN 8 (2009), 1433–1487. MR 2496770, DOI 10.1093/imrn/rnn165
- Javier Parcet, Pseudo-localization of singular integrals and noncommutative Calderón-Zygmund theory, J. Funct. Anal. 256 (2009), no. 2, 509–593. MR 2476951, DOI 10.1016/j.jfa.2008.04.007
- José Ángel Peláez, Jouni Rättyä, and Brett D. Wick, Bergman projection induced by kernel with integral representation, J. Anal. Math. 138 (2019), no. 1, 325–360. MR 3996042, DOI 10.1007/s11854-019-0035-5
- D. H. Phong and E. M. Stein, Estimates for the Bergman and Szegö projections on strongly pseudo-convex domains, Duke Math. J. 44 (1977), no. 3, 695–704. MR 450623, DOI 10.1215/S0012-7094-77-04429-5
- Gilles Pisier and Quanhua Xu, Non-commutative $L^p$-spaces, Handbook of the geometry of Banach spaces, Vol. 2, North-Holland, Amsterdam, 2003, pp. 1459–1517. MR 1999201, DOI 10.1016/S1874-5849(03)80041-4
- Narcisse Randrianantoanina, Non-commutative martingale transforms, J. Funct. Anal. 194 (2002), no. 1, 181–212. MR 1929141
- C. Stockdale, N. Wagner, Weighted endpoint bounds for the Bergman and Cauchy-Szegő projections on domains with near minimal smoothness. arXiv:2005.12261, 2020.
- V. P. Zaharjuta and V. I. Judovič, The general form of a linear functional in $H_{p}^{\prime }$, Uspehi Mat. Nauk 19 (1964), no. 2(116), 139–142 (Russian). MR 185424
- Ke He Zhu, Operator theory in function spaces, Monographs and Textbooks in Pure and Applied Mathematics, vol. 139, Marcel Dekker, Inc., New York, 1990. MR 1074007
Bibliographic Information
- Liang Wang
- Affiliation: School of Mathematics and Statistics, Wuhan University, Wuhan 430072, People’s Republic of China
- ORCID: 0000-0002-5046-7501
- Email: wlmath@whu.edu.cn
- Bang Xu
- Affiliation: Department of Mathematical Sciences, Seoul National University, Seoul 08826, Republic of Korea
- MR Author ID: 1288802
- Email: bangxu@snu.ac.kr
- Dejian Zhou
- Affiliation: School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha 410083, China
- MR Author ID: 1113565
- Email: zhoudejian@csu.edu.cn
- Received by editor(s): February 19, 2022
- Received by editor(s) in revised form: July 12, 2022
- Published electronically: December 21, 2022
- Additional Notes: The first author and second author are supported by Natural Science Foundation of China Grant: 12071355. The second author is also supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) Grant: NRF-2022R1A2C1092320 and the Samsung Science and Technology Foundation under Project Number: SSTF-BA2002-01. The third author is supported by Natural Science Foundation of China Grant: 12001541, Natural Science Foundation Hunan Grant: 2021JJ40714, Changsha Municipal Natural Science Foundation Grant: kq2014118.
The third author is the corresponding author. - Communicated by: Javad Mashreghi
- © Copyright 2022 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 151 (2023), 1221-1234
- MSC (2020): Primary 46L51, 42B20; Secondary 32A25, 46L52
- DOI: https://doi.org/10.1090/proc/16213
- MathSciNet review: 4531650