Bergman projection induced by radial weight acting on growth spaces

Abstract

Let \(\omega \) be a radial weight on the unit disc of the complex plane \(\mathbb {D}\) and denote by \(\widehat{\omega }(r)=\int _r^1 \omega (s)\,ds\) the tail integrals. A radial weight \(\omega \) belongs to the class \(\widehat{\mathcal {D}}\) if satisfies the upper doubling condition

$$\begin{aligned} \sup _{0<r<1}\frac{\widehat{\omega }(r)}{\widehat{\omega }\left( \frac{1+r}{2}\right) }<\infty . \end{aligned}$$

If \(\nu \) or \(\omega \) belongs to \(\widehat{\mathcal {D}}\), we describe the boundedness of the Bergman projection \(P_\omega \) induced by \(\omega \) on the growth space \(L^\infty _{\widehat{\nu }} =\{ f: \Vert f\Vert _{\infty ,v}=\text {ess sup}_{z\in \mathbb {D}} |f(z)|\widehat{\nu }(z)<\infty \}\) in terms of neat conditions on the moments and/or the tail integrals of \(\omega \) and \(\nu \). Moreover, we solve the analogous problem for \(P_\omega \) from \(L^\infty _{\widehat{\nu }}\) to the Bloch type space \(\mathcal {B}^\infty _{\widehat{\nu }} = \{f\, \text {analytic in} \mathbb {D}: \Vert f\Vert _{\mathcal {B}^\infty _{\widehat{\nu }}} \) \(= \sup _{z\in \mathbb {D}}(1-|z|)\widehat{\nu }(z)|f'(z)|<\infty \}.\) Similar questions for exponentially decreasing radial weights will also be studied.