Bergman projection induced by radial weight acting on growth spaces

Abstract

Let $ω$ be a radial weight on the unit disc of the complex plane $\mathbb{D}$ and denote $ω_x =\int_0^1 s^x ω(s)\,ds$, $x\ge 0$, for the moments of $ω$ and $\widehatω(r)=\int_r^1 ω(s)\,ds$ for the tail integrals. A radial weight $ω$ belongs to the class $\widehat{\mathcal{D}}$ if satisfies the upper doubling condition $$\sup_{0<r<1}\frac{\widehatω(r)}{\widehatω\left(\frac{1+r}{2}\right)}<\infty.$$ If $ν$ or $ω$ belongs to $\widehat{\mathcal{D}}$, it is described the boundedness of the Bergman projection $P_ω$ induced by $ω$ on the growth space $L^\infty_{\widehatν} =\{ f: \|f\|_{\infty,v}={ esssup}_{z\in\mathbb{D}} |f(z)|\widehatν(z)<\infty\}$ in terms of neat conditions on the moments and/or the tail integrals of $ω$ and $ν$. Moreover, it is solved the analogous problem for $P_ω$ from $L^\infty_{\widehatν}$ to the Bloch type space $B^\infty_{\widehatν}$ of analytic functions such that $\sup_{z\in \mathbb{D}}(1-|z|)\widehatν(z) |f'(z)|<\infty.$ We also study similar questions for exponentially decreasing radial weights.

Theorems & Definitions (40)

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