Weak-type estimates for the Bergman projection on planar domains

Abstract

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