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Let $\Omega \subset \mathbb{C}$ be a bounded, simply connected domain. Wegeneralize well-known weighted inequalities due to B\'{e}koll\`{e} and Bonamion $\mathbb D$ by giving a sufficient condition on the domain $\Omega$ suchthat the Bergman projection $\Pi_{\Omega}$ is bounded on $L^p(\Omega, \sigma)$for $1<p<\infty$ and $\sigma$ belonging to the Muckenhoupt class$\mathrm{B}_p(\Omega)$ whose basis consists of the images, under the Riemannmap, of Carleson boxes in $\mathbb{D}$. We also show that in the special caseof uniform domains, the same condition is also necessary for the full-rangeweighted estimates, and moreover provide an alternative characterization of theweight class that is intrinsic to $\Omega$. Our approach uses techniques fromboth conformal mapping and dyadic harmonic analysis. We also apply thesemethods to prove endpoint weak-type estimates and limited-range weightedinequalities.

Weighted estimates for the Bergman projection on planar domains