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On Singular Integrals and Quantitative Rectifiability in Parabolic Space and the Heisenberg Group

John Hoffman, Ben Jaye

TL;DR

This work extends the David–Semmes program to non-Euclidean geometries by establishing a parabolic-space analogue and a Heisenberg-group analogue of the uniform rectifiability criterion tied to $L^2$-bounded Calderón–Zygmund operators on ADR measures. In the Heisenberg group, boundedness of all dimension-3 CZOs suffices to imply uniform rectifiability without extra hypotheses, while in the parabolic setting an additionalHigher Spatial Dimension Condition (HSDC) is necessary to rule out vampiric counterexamples and recover a DS-type result via an alpha-number–driven cylinder blow-up scheme. The authors develop a multiscale framework combining $oldsymbol{α}$-numbers, geometric Littlewood–Paley theory, and vampiric-measure analysis, producing a robust mechanism to pass from qualitative control to quantitative UR in both geometries. These results advance the non-Euclidean rectifiability theory and suggest a pathway toward Nazarov–Tolsa–Volberg–type theorems for caloric and parabolic measures, with potential implications for parabolic and sub-Riemannian analysis.

Abstract

David and Semmes proved that if all CZOs (of suitable dimension) are bounded with respect to an Ahlfors regular measure, then the measure is uniformly rectifiable. We extend this theorem to the parabolic space and the first Heisenberg group.

On Singular Integrals and Quantitative Rectifiability in Parabolic Space and the Heisenberg Group

TL;DR

This work extends the David–Semmes program to non-Euclidean geometries by establishing a parabolic-space analogue and a Heisenberg-group analogue of the uniform rectifiability criterion tied to -bounded Calderón–Zygmund operators on ADR measures. In the Heisenberg group, boundedness of all dimension-3 CZOs suffices to imply uniform rectifiability without extra hypotheses, while in the parabolic setting an additionalHigher Spatial Dimension Condition (HSDC) is necessary to rule out vampiric counterexamples and recover a DS-type result via an alpha-number–driven cylinder blow-up scheme. The authors develop a multiscale framework combining -numbers, geometric Littlewood–Paley theory, and vampiric-measure analysis, producing a robust mechanism to pass from qualitative control to quantitative UR in both geometries. These results advance the non-Euclidean rectifiability theory and suggest a pathway toward Nazarov–Tolsa–Volberg–type theorems for caloric and parabolic measures, with potential implications for parabolic and sub-Riemannian analysis.

Abstract

David and Semmes proved that if all CZOs (of suitable dimension) are bounded with respect to an Ahlfors regular measure, then the measure is uniformly rectifiable. We extend this theorem to the parabolic space and the first Heisenberg group.

Paper Structure

This paper contains 32 sections, 46 theorems, 165 equations.

Table of Contents

  1. Introduction
  2. Horizontally antisymmetric kernels
  3. Vertical $\beta$-numbers and uniform rectifiability
  4. The precise result in the parabolic case
  5. The proof scheme
  6. Acknowledgements
  7. Preliminaries and main results
  8. Balls, cubes and cylinders
  9. Lipschitz functions
  10. Measures
  11. Dyadic cubes
  12. Carleson Families
  13. The Geometric Littlewood-Paley Condition
  14. Hausdorff Measure
  15. Alpha Numbers
  16. ...and 17 more sections

Key Result

Theorem 1.1

Suppose that $\mu$ is a $k$-Ahlfors regular (henceforth $k$-ADR) for $k\in \{1,\dots, n-1\}$. Then every convolution singular integral operator associated with a kernel $K\in C^{\infty}(\mathbb{R}^n\backslash \{0\})$ satisfying is bounded on $L^2(\mu)$ if and only if $\mu$ is uniformly rectifiable.

Theorems & Definitions (73)

  • Theorem 1.1: David-Semmes
  • Theorem 1.2
  • Conjecture 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6: BHMN
  • Theorem 1.7: BHMN2
  • Theorem 1.8
  • Corollary 1.9
  • Lemma 2.1
  • ...and 63 more