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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.

MSC:

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
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References:

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