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 ${\omega }_x={\int }_0^1{s}^x\omega (s)ds$, $x\ge 0$, for the moments of $\omega$ and $\hat{\omega }(r)={\int }_r^1\omega (s)ds$ for the tail integrals. A radial weight $\omega$ belongs to the class $\hat{\mathcal{D}}$ if satisfies the upper doubling condition $$\underset{0<r<1}{\sup }\frac{\hat{\omega }(r)}{\hat{\omega }\left(\frac{1+r}{2}\right)}<\infty .$$ If $\nu$ or $\omega$ belongs to $\hat{\mathcal{D}}$, it is described the boundedness of the Bergman projection $P_{\omega }$ induced by $\omega$ on the growth space $L_{\hat{\nu }}^{\infty }=\{f:\Vert f{\Vert }_{\infty ,v}={esssup}_{z\in \mathbb{D}}|f(z)|\hat{\nu }(z)<\infty \}$ in terms of neat conditions on the moments and/or the tail integrals of $\omega$ and $\nu$. Moreover, it is solved the analogous problem for $P_{\omega }$ from $L_{\hat{\nu }}^{\infty }$ to the Bloch type space $B_{\hat{\nu }}^{\infty }$ of analytic functions such that ${\sup }_{z\in \mathbb{D}}(1-|z|)\hat{\nu }(z)|f^{\prime }(z)|<\infty .$ We also study similar questions for exponentially decreasing radial weights.