# Bergman projection induced by radial weight

## TL;DR

This paper characterizes when the radial weight-induced Bergman projection is bounded or surjective between specific spaces, solves longstanding problems, and provides criteria based on doubling conditions, advancing the understanding of weighted Bergman spaces.

## Contribution

It offers new characterizations of radial weights for boundedness and surjectivity of the Bergman projection, and solves key open problems in the theory of weighted Bergman spaces.

## Key findings

- Characterizations of weights for bounded and surjective Bergman projections.
- Solution to Dostanić's problem under weak regularity conditions.
- Criteria for Littlewood-Paley estimates and two-weight inequalities.

## Abstract

The question of when the Bergman projection $P_\omega$ induced by a radial weight $\omega$ on the unit disc is a bounded operator from one space into another is of primordial importance in the theory of Bergman spaces. The long-standing problem of describing the radial weights $\omega$ such that $P_\omega$ is bounded on the Lebesgue space $L^p_\omega$ had been known to experts since decades before it was formally posed by Dostani\'c in 2004. A natural limit case of this setting is when $P_\omega$ acts from $L^\infty$ to the Bloch space. The surjectivity of the operator becomes another relevant question in this limit case. The main findings of this study are shortly listed as follows. We establish characterizations of the radial weights $\omega$ on the unit disc such that $P_\omega:L^\infty\to\mathcal{B}$ is bounded and/or acts surjectively, or the dual of $A^1_\omega$ is isomorphic to the Bloch space $\mathcal{B}$ under the $A^2_\omega$-pairing. We also solve the problem posed by Dostani\'c under a weak regularity hypothesis on the weight involved. With regard to Littlewood-Paley estimates, we describe the radial weights $\omega$ such that the norm of any function in $A^p_\omega$ is comparable to the norm in $L^p_\omega$ of its derivative times the distance from the boundary. This last-mentioned result solves another well-known problem on the area. All characterizations can be given in terms of doubling conditions on moments and/or tail integrals $\int_r^1\omega(t)\,dt$ of $\omega$, and are therefore easy to interpret. We also make substantial progress about the two weight inequality $$ \|P_\omega(f)\|_{L^p_\nu}\le C\|f\|_{L^p_\nu},\quad f\in L^p_\nu, \quad 1<p<\infty. $$ for radial weights $\omega$ and $\nu$.