CR functions at CR singularities: approximation, extension, and hulls

TL;DR

This paper investigates different definitions of CR functions at singular points, explores their extension and approximation properties, and constructs disc hulls to facilitate these extensions, highlighting distinctions among various classes.

Contribution

It introduces new methods for extending and approximating CR functions at singularities using disc hulls and provides a comprehensive comparison of different CR function classes.

Findings

Disc hulls enable fixed-neighborhood extension and approximation.

Distinct classes of CR functions exhibit different extension properties.

Examples illustrate the differences between classes and properties studied.

Abstract

We study three possible definitions of the notion of CR functions at CR singular points, their extension to a fixed-neighborhood of the singular point, and analogues of the Baouendi--Tr\`eves approximation in a fixed neighborhood. In particular, we give a construction of certain disc hulls, which, if large enough, give the fixed-neighborhood extension and approximation properties. We provide many examples showing the distinctions between the classes and the various properties studied.