Weak-type estimates for the Bergman projection on planar domains

We investigate the relationship between the weak-type regularity of the Bergman projection, ${\mathrm{\Pi }}_{\mathrm{\Omega }}$, of a simply connected domain $\mathrm{\Omega }\subset \mathbb{C}$ and the boundary geometry of $\mathrm{\Omega }$ in terms of a conformal map $\psi \text{\hspace{0.17em}⁣}:\mathbb{D}\to \mathrm{\Omega }$. We show that ${\mathrm{\Pi }}_{\mathrm{\Omega }}$ is of weak-type $(1,1)$ whenever $\mathrm{\mid }{\psi }^{\mathrm{\prime }}\mathrm{\mid }$ is in the Bekollé-Bonami class $B_1$, give a more general necessary condition for the weak-type $(p,p)$ bounds of ${\mathrm{\Pi }}_{\mathrm{\Omega }}$ when $1\le p<\mathrm{\infty }$, 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 ${\mathrm{\Pi }}_{\mathbb{D}}$. We provide several applications.