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General fractional derivatives and the Bergman projection. (English) Zbl 1457.32008

A radial weight \(\omega(z)=\omega(|z|)\) on \(\mathbb{D}\) is in the class \(\widehat{\mathcal D }\) if there exists \(C_\omega > 0\) so that for any \(0\leq r<1\), \[\widehat{\omega}(r):=\int_1^1 \omega(s)ds\leq C_\omega \widehat{\omega}\left(\frac{1+r}2\right).\]
Furthermore, \(\omega\in \widehat{\mathcal D}\), if there exist \(K_\omega > 1\) and \(C'_\omega>1\), such that
\[\widehat{\omega} (r)\geq C'_\omega\, \widehat{\omega}\left( 1-\frac{1- r}{K_\omega} \right).\]
A radial weight \(\omega\) is a doubling weight (\(\omega\in \mathcal{D}\)) if \(\omega\in\widehat{\mathcal D }\cap \widehat{\mathcal D}\).
If \(\omega\) is a doubling weight, \(A_\omega^p\) is the weighted Bergman space of holomorphic functions \(f\) on \(\mathbb D\) such that \[\|f\|_{A_\omega^p}=\int_{\mathbb D} |f(\xi)|^p\omega(\xi)dA(\xi)<\infty.\] If \(B_z^\omega\) is the reproducing Bergman kernel induced by the doubling weight \(\omega\), the associated integral operator is denoted by \(P_\omega\).
The purpose of this paper is the study of general fractional derivatives induced by weighted Bergman kernels. Given two radial weights, \(\omega,\nu\), there is a unique mapping \(R^{\omega,\nu}\), which is defined for all analytic functions and satisfies \(R^{\omega,\nu}(B_z^\omega)(\xi)= B_z^\nu(\xi)\).
J. Á. Peláez and J. Rättyä [“Bergman projection induced by radial weight”, Preprint, arXiv:1902.09837] have proved the surjectivity of the Bergman projection \(P^\omega\) from \(L^\infty\) to the Bloch space \({\mathcal B}\), where \({\mathcal B}\) is the space of holomorphic functions on \(\mathbb D\) such that \((1-|z|^2)|f'(z)|+|f'(0)|<+\infty\). In this paper a new proof of this surjectivity is obtained, using the general fractional derivatives \(R^{\omega,\eta}\):
Theorem. Let \(\omega\) be a doubling weight. Then the Bergman projection \(P_\omega:L^\infty\to {\mathcal B} \) is surjective. Moreover if \(\alpha>0\) and \(h\in {\mathcal B}\), then \[g_\alpha(z)= (1-z)^\alpha R^{\omega,\omega_\alpha}h(z)\in L^\infty,\quad P^\omega(g_\alpha)=h,\] where \(\omega_\alpha(z)=(1-z)^\alpha\omega(z)\).
The same method can be used to construct several pre-images of a given Bloch function.

MSC:

32A36 Bergman spaces of functions in several complex variables
30H30 Bloch spaces
30H25 Besov spaces and \(Q_p\)-spaces
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References:

[1] Peláez, J. Á.,A. Perälä, andJ. Rättyä: Hankel operators induced by radial Bekollé- Bonami weights on Bergman spaces. - Math. Zeit. (to appear), https://doi.org/10.1007/s00209019-02412-8.
[2] Peláez, J. Á., andJ. Rättyä: Weighted Bergman spaces induced by rapidly increasing weights. - Mem. Amer. Math. Soc. 1066:iii-vi, 2014. · Zbl 1308.30001
[3] Peláez, J. Á., andJ. Rättyä: Two weight inequality for Bergman projection. - J. Math. Pures Appl. 105, 2016, 102-130. · Zbl 1337.30064
[4] Peláez, J. Á., andJ. Rättyä: Bergman projection induced by radial weight. - Preprint. General fractional derivatives and the Bergman projection913
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