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

TL;DR

This paper establishes weighted endpoint bounds for the Bergman and Cauchy-Szeg ext{"o} projections on domains with near minimal smoothness, extending their weak-type properties under specific geometric and weight conditions.

Contribution

It proves weak-type (1,1) bounds for these projections on less smooth domains with weighted measures, a significant extension of previous results.

Findings

Weak-type (1,1) property for Bergman projection on strongly pseudoconvex domains with $C^4$ boundary.

Weak-type (1,1) property for Cauchy-Szeg ext{"o} projection on domains with $C^3$ boundary.

Weighted Kolmogorov and Zygmund inequalities for the projections.

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.