Nonabelian ramified coverings and $L^p$-boundedness of Bergman projections in $\mathbb C^2$

TL;DR

This paper investigates the $L^p$-boundedness of Bergman projections on complex domains covered by well-understood domains via ramified coverings, especially those with finite unitary reflection groups, establishing boundedness for all p in (1,∞).

Contribution

It demonstrates $L^p$-boundedness of Bergman projections for a broad class of ramified coverings with finite unitary reflection groups, extending known results to new complex domain classes.

Findings

Proves $L^p$-boundedness for all p in (1,∞) in specific ramified coverings.

Identifies conditions under which Bergman projections are bounded in these settings.

Extends classical results to complex domains with ramified coverings by reflection groups.

Abstract

In this work we explore the theme of $L^p$-boundedness of Bergman projections of domains that can be covered, in the sense of ramified coverings, by "nice" domains (e.g. strictly pseudoconvex domains with real analytic boundary). In particular, we focus on two-dimensional normal ramified coverings whose covering group is a finite unitary reflection group. In an infinite family of examples we are able to prove $L^p$-boundedness of the Bergman projection for every $p\in(1,\infty)$.