Weighted endpoint bounds for the Bergman and Cauchy-Szeg\H{o} projections on domains with near minimal smoothness

Abstract

We study the Bergman projection, $\mathcal{B}$, and the Cauchy-Szeg\H{o} projection, $\mathcal{S}$, on bounded domains with near minimal smoothness. We prove that $\mathcal{B}$ has the weak-type $(1,1)$ property with respect to weighted measures assuming that the underlying domain is strongly pseudoconvex with $C^4$ boundary and the weight satisfies the $B_1$ condition, and the same property for $\mathcal{S}$ on domains with $C^3$ boundaries and weights satisfying the $A_1$ condition. We also obtain weighted Kolmogorov and weighted Zygmund inequalities for $\mathcal{B}$ and $\mathcal{S}$ in their respective settings as corollaries.