Weak-type estimates for the Bergman projection on the polydisc and the Hartogs triangle

TL;DR

This paper studies the weak-type regularity of the Bergman projection on the polydisc and Hartogs triangle, establishing new endpoint estimates and clarifying the operator's behavior at critical points.

Contribution

It proves the weak-type $L ext{log}L$ estimate for the polydisc and determines the weak-type $(4,4)$ behavior for the Hartogs triangle, resolving key endpoint questions.

Findings

Weak-type $L ext{log}L$ estimate for polydisc

Weak-type $(4,4)$ for Hartogs triangle

Non-weak-type $(rac{4}{3}, rac{4}{3})$ for Hartogs triangle

Abstract

In this paper, we investigate the weak-type regularity of the Bergman projection. The two domains we focus on are the polydisc and the Hartogs triangle. For the polydisc we provide a proof that the weak-type behavior is of "$L\log L$" type. This result is likely known to the experts, but does not appear to be in the literature. For the Hartogs triangle we show that the operator is of weak-type $(4,4)$; settling the question of the behavior of the projection at this endpoint. At the other endpoint of interest, we show that the Bergman projection is not of weak-type $(\frac{4}{3}, \frac{4}{3})$ and provide evidence as to what the correct behavior at this endpoint might be.