Weak-type estimates for the Bergman projection on planar domains

We investigate the relationship between the weak-type regularity of the Bergman projection, $\Pi_{\Omega}$, of a simply connected domain $\Omega \subset \mathbb{C}$ and the boundary geometry of $\Omega$ in terms of a conformal map $\psi\colon\mathbb{D}\rightarrow\Omega$. We show that $\Pi_{\Omega}$ is of weak-type $(1,1)$ whenever $|\psi'|$ is in the Bekoll\'e-Bonami class $B_1$, give a more general necessary condition for the weak-type $(p,p)$ bounds of $\Pi_{\Omega}$ when $1\leq p<\infty$, and establish sharpened sufficient conditions for the weak-type bounds when $p>1$. Our results follow from a reformulation in terms of mixed-weighted weak-type inequalities for $\Pi_{\mathbb{D}}$. We provide several applications.