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Curvature, Hodge-Dirac operators and Riesz transforms

Cédric Arhancet

TL;DR

This work develops an abstract curvature-intertwining framework for Hodge–Dirac operators associated with semigroups on Banach spaces, linking lower Ricci-type curvature bounds to the boundedness of H^∞ functional calculi and Riesz transforms. Central to the approach are a pair of compatible gradient/divergence operators (∂,∂^†) for a sectorial generator A and a curvature condition Curv_∂(λ) (and its H^∞-enhanced variant) expressed via a tangential semigroup on a “tangent space.” Under suitable Riesz equivalences and curvature assumptions, the authors prove bisectoriality and bounded H^∞ calculus for the Hodge–Dirac operator D on the reduced space, and extend these results to the full space via a bounded projection Q, yielding R-bisectoriality and functional calculus for the full operator. The paper also derives an abstract L^p Poincaré inequality and showcases the framework in diverse settings, including Riemannian manifolds, generalized Ornstein–Uhlenbeck semigroups, q-OU semigroups, compact (quantum) groups, Schur multipliers, quantum tori, and subelliptic Laplacians on compact Lie groups. These results unify and extend known Riesz-transform and functional-calculus outcomes, and provide a robust tool for Banach-spectral triples and noncommutative L^p-analysis.

Abstract

We introduce a notion of Ricci curvature lower bound for strongly continuous semigroups of operators acting on a reflexive Banach space, endowed with a suitable pair of divergence and gradient operators. We use this notion to investigate the functional calculus of the Hodge-Dirac operator associated to the semigroup in link with the boundedness of suitable Riesz transforms. Our paper offers a unified framework that not only encapsulates existing results in some contexts but also yields new findings in others. This is demonstrated through applications in the frameworks of Riemannian manifolds, compact (quantum) groups, noncommutative tori, Ornstein-Uhlenbeck semigroup, $q$-Ornstein-Uhlenbeck semigroups and semigroups of Schur multipliers. We also provide an $\mathrm{L}^p$-Poincaré inequality that is applicable to all previously discussed contexts under suitable some assumptions of boundedness of functional calculus and Riesz transforms (and some uniform exponential stability). Finally, we prove the boundedness of some Riesz transforms in some contexts as compact Lie groups.

Curvature, Hodge-Dirac operators and Riesz transforms

TL;DR

This work develops an abstract curvature-intertwining framework for Hodge–Dirac operators associated with semigroups on Banach spaces, linking lower Ricci-type curvature bounds to the boundedness of H^∞ functional calculi and Riesz transforms. Central to the approach are a pair of compatible gradient/divergence operators (∂,∂^†) for a sectorial generator A and a curvature condition Curv_∂(λ) (and its H^∞-enhanced variant) expressed via a tangential semigroup on a “tangent space.” Under suitable Riesz equivalences and curvature assumptions, the authors prove bisectoriality and bounded H^∞ calculus for the Hodge–Dirac operator D on the reduced space, and extend these results to the full space via a bounded projection Q, yielding R-bisectoriality and functional calculus for the full operator. The paper also derives an abstract L^p Poincaré inequality and showcases the framework in diverse settings, including Riemannian manifolds, generalized Ornstein–Uhlenbeck semigroups, q-OU semigroups, compact (quantum) groups, Schur multipliers, quantum tori, and subelliptic Laplacians on compact Lie groups. These results unify and extend known Riesz-transform and functional-calculus outcomes, and provide a robust tool for Banach-spectral triples and noncommutative L^p-analysis.

Abstract

We introduce a notion of Ricci curvature lower bound for strongly continuous semigroups of operators acting on a reflexive Banach space, endowed with a suitable pair of divergence and gradient operators. We use this notion to investigate the functional calculus of the Hodge-Dirac operator associated to the semigroup in link with the boundedness of suitable Riesz transforms. Our paper offers a unified framework that not only encapsulates existing results in some contexts but also yields new findings in others. This is demonstrated through applications in the frameworks of Riemannian manifolds, compact (quantum) groups, noncommutative tori, Ornstein-Uhlenbeck semigroup, -Ornstein-Uhlenbeck semigroups and semigroups of Schur multipliers. We also provide an -Poincaré inequality that is applicable to all previously discussed contexts under suitable some assumptions of boundedness of functional calculus and Riesz transforms (and some uniform exponential stability). Finally, we prove the boundedness of some Riesz transforms in some contexts as compact Lie groups.
Paper Structure (40 sections, 36 theorems, 262 equations, 2 figures)

This paper contains 40 sections, 36 theorems, 262 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Structure of the Paper
  3. Preliminaries
  4. Operator theory
  5. Unbounded operators
  6. Semigroup theory
  7. Sectorial operators
  8. Fractional powers
  9. Bisectorial operators
  10. Functional calculus
  11. Functional calculus of Hodge-Dirac operators
  12. Commutation relations
  13. Riesz equivalences
  14. Pairs of divergence and gradient operators
  15. Curvature via commutation relations and bounded $\mathrm{H}^\infty$ functional calculus
  16. ...and 25 more sections

Key Result

Proposition 2.1

Let $R \colon X \to Y$ a bounded operator and let $S \colon Y \to Z$ be a closable operator with $\mathop{\mathrm{Ran}}\nolimits R \subset \mathop{\mathrm{dom}}\nolimits S$. Then $SR \colon X \to Z$ is bounded.

Figures (2)

  • Figure 1: open sector $\Sigma_\theta$ of angle $2\theta$.
  • Figure 2: the spectrum $\sigma(D)$ of a bisectorial operator $D$

Theorems & Definitions (57)

  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Example 2.4
  • Example 2.5
  • Example 2.6
  • Proposition 2.7
  • Proposition 3.1
  • Remark 3.2
  • Definition 3.3
  • ...and 47 more