Bergman projection induced by radial weight
José Ángel Peláez, Jouni Rättyä
TL;DR
This work provides a comprehensive characterization of radial weights ω on the unit disk that govern the boundedness and surjectivity of Bergman projections P_ω across a spectrum of spaces, including L^∞→Bloch and L^p_ω→L^p_ω. Central techniques rely on moment analysis via ω_{2n+1} and tail integrals ω̂, enabling equivocal criteria organized into weight classes ẊD, 𝓜, and 𝒟 and yielding sharp Littlewood-Paley formulas. The authors establish duality identifications such as (A^1_ω)^* ≅ BMOA(∞,ω) and characterize two-weight inequalities for P_ω through new functionals A_p(ω,ν) and M_p(ω,ν), clarifying when a single kernel governs mappings between weighted spaces. They also connect these operator-theoretic questions to structural properties of ω via doubling-type conditions on moments, providing practical criteria with implications for both one- and two-weight Bergman-projection problems. Overall, the paper unifies the operator theory of Bergman spaces under a radial-weight framework, advancing understanding of projection boundedness, duality, and Littlewood-Paley phenomena.
Abstract
The question of when the Bergman projection $P_ω$ induced by a radial weight $ω$ 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 $ω$ such that $P_ω$ is bounded on the Lebesgue space $L^p_ω$ had been known to experts since decades before it was formally posed by Dostanić in 2004. A natural limit case of this setting is when $P_ω$ 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 $ω$ on the unit disc such that $P_ω:L^\infty\to\mathcal{B}$ is bounded and/or acts surjectively, or the dual of $A^1_ω$ is isomorphic to the Bloch space $\mathcal{B}$ under the $A^2_ω$-pairing. We also solve the problem posed by Dostanić under a weak regularity hypothesis on the weight involved. With regard to Littlewood-Paley estimates, we describe the radial weights $ω$ such that the norm of any function in $A^p_ω$ is comparable to the norm in $L^p_ω$ 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 $\int_r^1ω(t)\,dt$ of $ω$, and are therefore easy to interpret. We also make substantial progress about the two weight inequality $$ \|P_ω(f)\|_{L^p_ν}\le C\|f\|_{L^p_ν},\quad f\in L^p_ν, \quad 1<p<\infty. $$ for radial weights $ω$ and $ν$.