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

Jiri Lebl, Alan Noell, Sivaguru Ravisankar

TL;DR

The paper tackles the challenge of defining and extending CR functions at CR singular points by introducing and comparing three natural classes—CR, CR_P, and CR_H—and by developing disc-hull machinery (DH and SADH) to study extension and approximation in fixed neighborhoods. It establishes when fixed-neighborhood extension/approximation can be achieved via disc hulls, and it constructs explicit examples (notably flat elliptic and hyperbolic/parabolic Bishop surfaces, and a Levi-flat-type manifold) to separate the behaviors of the three function classes. Key contributions include proving fixed-neighborhood approximation under certain hull conditions, demonstrating that CR_P^ω can coincide with CR_H under strong iterated SADH, and showing that CR, CR_P, and CR_H can diverge in CR singular settings with concrete counterexamples. The results extend Baouendi–Trèves-type approximation into CR singular contexts, reveal that hulls can be large even when Levi-flat, and provide integral-operator constructions for specific surfaces, thereby clarifying the landscape of CR function extensions and polynomial approximations near CR singularities.

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--Trèves 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.

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

TL;DR

The paper tackles the challenge of defining and extending CR functions at CR singular points by introducing and comparing three natural classes—CR, CR_P, and CR_H—and by developing disc-hull machinery (DH and SADH) to study extension and approximation in fixed neighborhoods. It establishes when fixed-neighborhood extension/approximation can be achieved via disc hulls, and it constructs explicit examples (notably flat elliptic and hyperbolic/parabolic Bishop surfaces, and a Levi-flat-type manifold) to separate the behaviors of the three function classes. Key contributions include proving fixed-neighborhood approximation under certain hull conditions, demonstrating that CR_P^ω can coincide with CR_H under strong iterated SADH, and showing that CR, CR_P, and CR_H can diverge in CR singular settings with concrete counterexamples. The results extend Baouendi–Trèves-type approximation into CR singular contexts, reveal that hulls can be large even when Levi-flat, and provide integral-operator constructions for specific surfaces, thereby clarifying the landscape of CR function extensions and polynomial approximations near CR singularities.

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--Trèves 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.
Paper Structure (9 sections, 19 theorems, 35 equations)

This paper contains 9 sections, 19 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. CR functions
  3. Extension, approximation, and hulls
  4. Results on hulls and approximations
  5. Flat elliptic Bishop surfaces
  6. Baouendi--Trèves for a special elliptic Bishop surface
  7. Flat hyperbolic or parabolic Bishop surfaces
  8. A manifold with a large hull
  9. Fixed-neighborhood approximation without fixed-neighborhood extension

Key Result

Theorem 2.3

Let $M \subset {\mathbb{C}}^n$ be a real submanifold of regularity $C^\ell$ for $\ell \geq 1$.

Theorems & Definitions (53)

  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3
  • proof
  • Lemma 2.4
  • proof
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • ...and 43 more