The $L^p$- regularity problem for the Bergman projection of two-dimensional Rudin ball quotients

TL;DR

This paper resolves the $L^p$-regularity problem for the Bergman projection on two-dimensional Rudin ball quotients by lifting the problem along ramified coverings $\pi: B_2 \to D$ associated with a finite unitary reflection group $G$. The authors reduce the question to a kernel bound for $K_{G,p}$, derived from the jetting of $\pi$ and the Bergman kernel on the ball, and prove a dimension-two estimate $|K_{G,p}(z,w)| \le c \sum_{g\in G}|K_{B_2}(g.z,w)|$. Consequently, for every Rudin ball quotient $D$ in $\mathbb{C}^2$, the Bergman projection is bounded on $L^p$ for all $p\in(1,\infty)$, while $p=1$ or $p=\infty$ are excluded. The work highlights how the arrangement of reflecting hyperplanes in $G$ governs the analytic regularity of the Bergman projection on quotient domains, and provides a complete solution in this 2D setting.

Abstract

We solve the $L^p$-regularity problem of the Bergman projection of two-dimensional Rudin ball quotients.

Theorems & Definitions (31)

• Proposition 1.1: rudin_refl, DM
• Example 1.2: lehrer_taylor
• Proposition 1.3
• Theorem 1.4
• Corollary 1.5
• Proposition 1.6: lehrer_taylor
• Lemma 2.1
• proof
• Lemma 2.2
• proof