On the weighted estimate of the Bergman projection. (English) Zbl 1487.47057

The weighted Bergman space \(A_\alpha^p(\mathcal{H})\) on the upper-half plane \(\mathcal{H}:=\{z=x+iy\in\mathbb{C}: y>0\}\) for \(-1<\alpha<1\), \(1<p<\infty\), consists of all analytic functions \(f\) on \(\mathcal{H}\) such that \[\|f\|^p_{p,\alpha}=\int_\mathcal{H}|f(x+iy)|^py^\alpha\,dx\,dy<\infty.\] For any \(f\in A_\alpha^p(\mathcal{H})\) holds \[f(w)=P_\alpha f(w)=\int_\mathcal{H}\frac{f(z)}{(z-\overline{w})^{2+\alpha}}y^\alpha\,dx\,dy<\infty,\quad w\in\mathcal{H}.\tag{1}\]
Let \(B_{p,\alpha}(\mathcal{H})\) denote the set of Békollé-Bonami non-negative weight functions: \[[\omega]_{B_{p,\alpha(\mathcal{H})}}=\sup\limits_{I\subset\mathbb{R}}\left(\frac1{|I|^{2+\alpha}}\int_{Q_I}\omega(z)y^\alpha\,dx\,dy\right)\left(\frac1{|I|^{2+\alpha}}\int_{Q_I}[\omega(z)]^{1-q}y^\alpha\,dx\,dy\right)^{p-1},\tag{2}\] where \(pq=p+q\), \(I\) is a subinterval of \(\mathbb{R}\) and \(Q_I:=\{z=x+iy\in\mathbb{C}: 0<y<|I|\}\).
Békollé-Bonami proved the boundedness of the Bergman projection \(P_\alpha\) in the weighted space \(L^p(\mathcal{H},\omega y^\alpha\,dx\,dy)\) [D. Bekollé, Stud. Math. 71, 305–323 (1982; Zbl 0516.47016); D. Békollé and A. Bonami, C. R. Acad. Sci., Paris, Sér. A 286, 775–778 (1978; Zbl 0398.30006)], while S. Pott and M. C. Reguera [J. Funct. Anal. 265, No. 12, 3233–3244 (2013; Zbl 1295.46020)] proved the estimate \[\|P_\alpha\varphi\big|L^p(\mathcal{H},\omega y^\alpha\,dx\,dy)\|\leqslant C_p[\omega]^{\max\{1,q/p\}}_{B_{p,\alpha}}\|\varphi\big|L^p(\mathcal{H},\omega y^\alpha\,dx\,dy)\|,\tag{3}\] where \(p,q\) are the same as above.
The author revisits the proof of the estimate (3) and provides a kind of simplification by avoiding the use of extrapolation.


47B38 Linear operators on function spaces (general)
30H20 Bergman spaces and Fock spaces
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
42A61 Probabilistic methods for one variable harmonic analysis
Full Text: DOI


[1] Aleman, A.; Pott, S.; Reguera, M. C., Sarason conjecture on the Bergman space, Int. Math. Res. Not. 2017 (2017), 4320-4349 · Zbl 1405.30055
[2] Bekollé, D., Inégalité à poids pour le projecteur de Bergman dans la boule unité de \(\mathbb C^n\), Stud. Math. 71 (1982), 305-323 French · Zbl 0516.47016
[3] Bekollé, D.; Bonami, A., Inégalités à poids pour le noyau de Bergman, C. R. Acad. Sci., Paris, Sér. A 286 (1978), 775-778 French · Zbl 0398.30006
[4] Cruz-Uribe, D., The invertibility of the product of unbounded Toeplitz operators, Integral Equations Oper. Theory 20 (1994), 231-237 · Zbl 0817.47034
[5] García-Cuerva, J.; Francia, J. L. Rubio de, Weighted Norm Inequalities and Related Topics, North-Holland Mathematics Studies, 116 Notas de Matemática (104), North-Holland Publishing, Amsterdam (1985) · Zbl 0578.46046
[6] Hytönen, T. P.; Lacey, M. T.; Pérez, C., Sharp weighted bounds for the \(q\)-variation of singular integrals, Bull. Lond. Math. Soc. 45 (2013), 529-540 · Zbl 1271.42021
[7] Hytönen, T.; Pérez, C., Sharp weighted bounds involving \(A_{∞}\), Anal. PDE 6 (2013), 777-818 · Zbl 1283.42032
[8] Isralowitz, J., Invertible Toeplitz products, weighted norm inequalities, and \( A_p\) weights, J. Oper. Theory 71 (2014), 381-410 · Zbl 1313.47059
[9] Lerner, A. K., A simple proof of the \(A_2\) conjecture, Int. Math. Res. Not. 2013 (2013), 3159-3170 · Zbl 1318.42018
[10] Michalska, M.; Nowak, M.; Sobolewski, P., Bounded Toeplitz and Hankel products on weighted Bergman spaces of the unit ball, Ann. Pol. Math. 99 (2010), 45-53 · Zbl 1239.47021
[11] Moen, K., Sharp weighted bounds without testing or extrapolation, Arch. Math. 99 (2012), 457-466 · Zbl 1266.42037
[12] Nazarov, F., A counterexample to Sarason’s conjecture, Preprint available at http://users.math.msu.edu/users/fedja/Preprints/Sarps.html
[13] Pott, S.; Reguera, M. C., Sharp Békollé estimates for the Bergman projection, J. Funct. Anal. 265 (2013), 3233-3244 · Zbl 1295.46020
[14] Pott, S.; Strouse, E., Products of Toeplitz operators on the Bergman spaces \(A^2_α\), Algebra Anal. 18 (2006), 144-161 translation in St. Petersbg. Math. J. 18 2007 105-118 · Zbl 1127.47028
[15] Sarason, D., Products of Toeplitz operators, Linear and Complex Analysis Problem Book 3, Part I V. P. Havin, N. K. Nikolski Lecture Notes in Mathematics 1573, Springer, Berlin (1994), 318-319 · Zbl 0893.30036
[16] Stroethoff, K.; Zheng, D., Bounded Toeplitz products on the Bergman space of the polydisk, J. Math. Anal. Appl. 278 (2003), 125-135 · Zbl 1051.47025
[17] Stroethoff, K.; Zheng, D., Bounded Toeplitz products on Bergman spaces of the unit ball, J. Math. Anal. Appl. 325 (2007), 114-129 · Zbl 1111.32003
[18] Stroethoff, K.; Zheng, D., Bounded Toeplitz products on weighted Bergman spaces, J. Oper. Theory 59 (2008), 277-308 · Zbl 1199.47127
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.