Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Analytic holonomicity of real C$^{\mathrm{exp}}$-class distributions

Avraham Aizenbud, Raf Cluckers, Michel Raibaut, Tamara Servi

TL;DR

This work introduces distributions of ${\mathscr C}^{\mathrm{exp}}$-class on open subsets of ${\mathbb{R}}^n$ via continuous wavelet transforms and builds on the ${\mathscr C}^{\mathrm{exp}}$-class framework from CCMRS to establish real-analytic holonomicity for all such distributions. The authors prove stability under push-forward, pull-back, derivation, and anti-derivation, and show tempered ${\mathscr C}^{\mathrm{exp}}$-class distributions retain their class under Fourier transform. A key novelty is the use of anti-derivatives and a refined resolution of subanalytic functions to reduce to locally integrable cases, enabling a full holonomicity result in the archimedean setting. The paper also develops an algebraic-holonomicity perspective, provides a resolution theorem for subanalytic sets, and discusses generalizations to broader ${\mathcal{C}}^{\mathbb C,{\mathcal F}}$-class frameworks, while outlining open questions about temperedness and wave-front structure. Together, these results place ${\mathscr C}^{\mathrm{exp}}$-class distributions among the robust holonomic toolkit of real-analytic D-modules with broad applicability to wavelet-analytic methods and subanalytic geometry.

Abstract

We introduce a notion of distributions on $\mathbb{R}^n$, called distributions of C$^{\mathrm{exp}}$-class, based on wavelet transforms of distributions and the theory from Cluckers, Comte, Miller, Rolin, Servi (2018) about C$^{\mathrm{exp}}$-class functions. We prove that the framework of C$^{\mathrm{exp}}$-class distributions is closed under natural operations, like push-forward, pull-back, derivation and anti-derivation, and, in the tempered case, Fourier transforms. Our main result is the (real analytic) holonomicity of all distributions of C$^{\mathrm{exp}}$-class.

Analytic holonomicity of real C$^{\mathrm{exp}}$-class distributions

TL;DR

This work introduces distributions of -class on open subsets of via continuous wavelet transforms and builds on the -class framework from CCMRS to establish real-analytic holonomicity for all such distributions. The authors prove stability under push-forward, pull-back, derivation, and anti-derivation, and show tempered -class distributions retain their class under Fourier transform. A key novelty is the use of anti-derivatives and a refined resolution of subanalytic functions to reduce to locally integrable cases, enabling a full holonomicity result in the archimedean setting. The paper also develops an algebraic-holonomicity perspective, provides a resolution theorem for subanalytic sets, and discusses generalizations to broader -class frameworks, while outlining open questions about temperedness and wave-front structure. Together, these results place -class distributions among the robust holonomic toolkit of real-analytic D-modules with broad applicability to wavelet-analytic methods and subanalytic geometry.

Abstract

We introduce a notion of distributions on , called distributions of C-class, based on wavelet transforms of distributions and the theory from Cluckers, Comte, Miller, Rolin, Servi (2018) about C-class functions. We prove that the framework of C-class distributions is closed under natural operations, like push-forward, pull-back, derivation and anti-derivation, and, in the tempered case, Fourier transforms. Our main result is the (real analytic) holonomicity of all distributions of C-class.
Paper Structure (16 sections, 51 theorems, 86 equations)

This paper contains 16 sections, 51 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Motivation and context
  3. Sketch of the proof and difference with the non-archimedean case
  4. Functions of ${\mathscr C}$-class and of ${\mathscr C}^{\mathrm{exp}}$-class
  5. D-modules and distributions
  6. Basic notation
  7. Holonomic distributions
  8. Algebraic Holonomicity
  9. Distributions of ${\mathscr C}^{\mathrm{exp}}$-class
  10. Anti-derivatives
  11. Construction of the anti-derivative
  12. Anti-derivatives can make distributions into continuous functions
  13. A resolution result for subanalytic sets and functions
  14. Holonomicity of ${\mathscr C}^{\mathrm{exp}}$-class distributions
  15. Generalizations
  16. ...and 1 more sections

Key Result

Theorem 1

Any ${\mathscr C}^{\mathrm{exp}}$-class distribution $u$ on an open set $U\subset {\mathbb R}^n$ is holonomic.

Theorems & Definitions (119)

  • Theorem 1: Proof in Section \ref{['sec:holon']}
  • Theorem 2: Proof in Section \ref{['sec:antider']}
  • Corollary 3: Proof in Section \ref{['sec:antider']}
  • Example 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4: Theorem 2.12 CCMRS
  • Remark 2.5
  • Lemma 2.6
  • ...and 109 more