Czechoslovak Mathematical Journal, first online, pp. 1-15


On the weighted estimate of the Bergman projection

Benoit Florent Sehba

Received October 27, 2016.   First published February 8, 2018.

Abstract:  We present a proof of the weighted estimate of the Bergman projection that does not use extrapolation results. This estimate is extended to product domains using an adapted definition of Békollé-Bonami weights in this setting. An application to bounded Toeplitz products is also given.
Keywords:  Bergman space; reproducing kernel; Toeplitz operator; Békollé-Bonami weight
Classification MSC:  47B38, 30H20, 47B35, 42C40, 42A61
DOI:  10.21136/CMJ.2018.0531-16

PDF available at:  Institute of Mathematics CAS

References:
[1] A. Aleman, S. Pott, M. C. Reguera: Sarason conjecture on the Bergman space. Int. Math. Res. Not. 2017 (2017), 4320-4349. DOI 10.1093/imrn/rnw134 | MR 3674172
[2] D. Bekollé: Inégalité à poids pour le projecteur de Bergman dans la boule unité de ${\bb C}^n$. Stud. Math. 71 (1982), 305-323. (In French.) DOI 10.4064/sm-71-3-305-323 | MR 0667319 | Zbl 0516.47016
[3] D. Bekollé, A. Bonami: Inégalités à poids pour le noyau de Bergman. C. R. Acad. Sci., Paris, Sér. A 286 (1978), 775-778. (In French.) MR 0497663 | Zbl 0398.30006
[4] D. Cruz-Uribe: The invertibility of the product of unbounded Toeplitz operators. Integral Equations Oper. Theory 20 (1994), 231-237. DOI 10.1007/BF01679672 | MR 1294717 | Zbl 0817.47034
[5] J. García-Cuerva, J. L. Rubio de Francia: Weighted Norm Inequalities and Related Topics. North-Holland Mathematics Studies, 116 Notas de Matemática (104), North-Holland Publishing, Amsterdam (1985). DOI 10.1016/s0304-0208(08)x7154-3 | MR 0807149 | Zbl 0578.46046
[6] T. P. Hytönen, M. T. Lacey, C. Pérez: Sharp weighted bounds for the $q$-variation of singular integrals. Bull. Lond. Math. Soc. 45 (2013), 529-540. DOI 10.1112/blms/bds114 | MR 3065022 | Zbl 1271.42021
[7] T. Hytönen, C. Pérez: Sharp weighted bounds involving $A_{\infty}$. Anal. PDE 6 (2013), 777-818. DOI 10.2140/apde.2013.6.777 | MR 3092729 | Zbl 1283.42032
[8] J. Isralowitz: Invertible Toeplitz products, weighted norm inequalities, and $ A_p$ weights. J. Oper. Theory 71 (2014), 381-410. DOI 10.7900/jot.2012apr10.1989 | MR 3214643 | Zbl 1313.47059
[9] A. K. Lerner: A simple proof of the $A_2$ conjecture. Int. Math. Res. Not. 2013 (2013), 3159-3170. DOI 10.1093/imrn/rns145 | MR 3085756 | Zbl 1318.42018
[10] M. Michalska, M. Nowak, P. Sobolewski: Bounded Toeplitz and Hankel products on weighted Bergman spaces of the unit ball. Ann. Pol. Math. 99 (2010), 45-53. DOI 10.4064/ap99-1-4 | MR 2660591 | Zbl 1239.47021
[11] K. Moen: Sharp weighted bounds without testing or extrapolation. Arch. Math. 99 (2012), 457-466. DOI 10.1007/s00013-012-0453-4 | MR 3000426 | Zbl 1266.42037
[12] F. Nazarov: A counterexample to Sarason's conjecture. Preprint available at http://users.math.msu.edu/users/fedja/Preprints/Sarps.html.
[13] S. Pott, M. C. Reguera: Sharp Békollé estimates for the Bergman projection. J. Funct. Anal. 265 (2013), 3233-3244. DOI 10.1016/j.jfa.2013.08.018 | MR 3110501 | Zbl 1295.46020
[14] S. Pott, E. Strouse: Products of Toeplitz operators on the Bergman spaces $A^2_\alpha$. Algebra Anal. 18 (2006), 144-161. (English. Russian original.); translation in St. Petersbg. Math. J. 18 (2007), 105-118. DOI 10.1090/S1061-0022-06-00945-9 | MR 2225216 | Zbl 1127.47028
[15] D. Sarason: Products of Toeplitz operators. Linear and Complex Analysis Problem Book 3, Part I V. (eds P. Havin, N. K. Nikolski). Lecture Notes in Mathematics 1573, Springer, Berlin (1994), 318-319. DOI 10.1007/BFb0100201 | MR 1334345 | Zbl 0893.30036
[16] K. Stroethoff, D. Zheng: Bounded Toeplitz products on the Bergman space of the polydisk. J. Math. Anal. Appl. 278 (2003), 125-135. DOI 10.1016/S0022-247X(02)00578-4 | MR 1963469 | Zbl 1051.47025
[17] K. Stroethoff, D. Zheng: Bounded Toeplitz products on Bergman spaces of the unit ball. J. Math. Anal. Appl. 325 (2007), 114-129. DOI 10.1016/j.jmaa.2006.01.009 | MR 2273032 | Zbl 1111.32003
[18] K. Stroethoff, D. Zheng: Bounded Toeplitz products on weighted Bergman spaces. J. Oper. Theory 59 (2008), 277-308. MR 2411047 | Zbl 1199.47127

Affiliations:   Benoit Florent Sehba, Department of Mathematics University of Ghana, P.O. Box LG 62, Legon, Accra, Ghana, e-mail: bfsehba@ug.edu.gh


 
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