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 \textbf{only} for $p=2$, or for \textbf{all} $p\in(1,\infty)$. \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 Pel\'aez and R\"atty\"a in 2021 and thereby settles a problem formally posed by Dostani\'c in 2004.