Weak-type estimates for the Bergman projection on planar domains

Abstract

We investigate the relationship between the weak-type regularity of the Bergman projection, $Π_Ω$, of a simply connected domain $Ω\subset \mathbb{C}$ and the boundary geometry of $Ω$ in terms of a conformal map $ψ\colon\mathbb{D}\rightarrowΩ$. We show that $Π_Ω$ is of weak-type $(1,1)$ whenever $|ψ'|$ is in the Bekollé-Bonami class $B_1$, give a more general necessary condition for the weak-type $(p,p)$ bounds of $Π_Ω$ 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 $Π_{\mathbb{D}}$. We provide several applications.