A proof of the Peláez--Rättyä conjecture

We characterize the $L^p$-boundedness of Bergman projections induced by radial weights on the unit disk, establishing the following dichotomy: \begin{center} the projection is bounded either $\mathbf{\text{only}}$ for $p=2$, or for $\mathbf{\text{all}}$ $p\in (1,\mathrm{\infty })$. \end{center} Moreover, the latter alternative holds if and only if the weight satisfies the one-sided doubling condition. This confirms a conjecture proposed by Peláez and Rättyä in 2021 and thereby settles a problem formally posed by Dostanić in 2004.