Bergman projection induced by radial weight acting on growth spaces

TL;DR

This paper characterizes the boundedness of the Bergman projection induced by radial weights on growth spaces and Bloch-type spaces, using conditions on moments and tail integrals for weights in the class D4.

Contribution

It provides necessary and sufficient conditions for the boundedness of the Bergman projection on growth and Bloch-type spaces for weights in the class D4, including exponentially decreasing weights.

Findings

Boundedness characterized by moments and tail integrals.

Conditions established for weights in D4 class.

Extension to exponentially decreasing weights.

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