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Some remarks on monogenity

Tove Dahn

TL;DR

The paper builds a comprehensive framework for representations of monogenic functions on very regular groups by weaving together distribution theory, spectral analysis, and geometric control of continuations. It introduces notions such as contraction domains, polar subgroups, normal models, and involution conditions to rigorously characterize monogenity, removable sets, and boundary mappings. Through exponential representations, approximative representations, and a detailed treatment of singularities and monodromy, it provides criteria for when monogenic continuations exist and how they behave under various regularizations. The resulting framework links algebraic and analytic structures to practical tools for boundary reduction, invariant selection, and stable representations with potential implications for multi-variable holomorphy and complex-analytic continuation on structured groups. The work thus offers a unified, technically detailed approach to understanding monogenicity and its propagation under transformations and boundary phenomena in a highly regular operator-theoretic setting.

Abstract

We discuss representations of monogenic functions over very regular groups.

Some remarks on monogenity

TL;DR

The paper builds a comprehensive framework for representations of monogenic functions on very regular groups by weaving together distribution theory, spectral analysis, and geometric control of continuations. It introduces notions such as contraction domains, polar subgroups, normal models, and involution conditions to rigorously characterize monogenity, removable sets, and boundary mappings. Through exponential representations, approximative representations, and a detailed treatment of singularities and monodromy, it provides criteria for when monogenic continuations exist and how they behave under various regularizations. The resulting framework links algebraic and analytic structures to practical tools for boundary reduction, invariant selection, and stable representations with potential implications for multi-variable holomorphy and complex-analytic continuation on structured groups. The work thus offers a unified, technically detailed approach to understanding monogenicity and its propagation under transformations and boundary phenomena in a highly regular operator-theoretic setting.

Abstract

We discuss representations of monogenic functions over very regular groups.

Paper Structure

This paper contains 24 sections, 6 theorems.

Table of Contents

  1. Summable distributions
  2. Nuclearity and continuous groups
  3. Monogenity
  4. Reduction to the boundary
  5. Partial monogenity
  6. Spectral theory
  7. Joint invariants
  8. Removable set
  9. Polar subgroup
  10. Contraction domains
  11. Normal surfaces
  12. A strict condition
  13. A strict subgroup
  14. The involution condition
  15. Resolution of sng
  16. ...and 9 more sections

Key Result

Lemma 4

When $d_{\Gamma} T(f) <C$ defines a neighborhood of $\Gamma=\{ d_{\Gamma}=0 \}$, where $d^{2}_{\Gamma}$ algebraic, then $\Gamma$ can be regarded as removable.

Theorems & Definitions (6)

  • Lemma 4
  • Lemma 5
  • Proposition 6
  • Lemma 7
  • Lemma 8
  • Proposition 9