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Intrinsic notion of principal symbol for the Heisenberg calculus

Raphael Ponge

TL;DR

This paper defines an intrinsic principal symbol for the hypoelliptic Heisenberg calculus on Heisenberg manifolds, realized as a homogeneous section $\sigma_m(P)$ of $S_m(\mathfrak{g}^{*}M,\mathcal{E})$ over the tangent Lie group bundle $GM$, linking tangent-group data with Rockland criteria and parametrix existence while clarifying the intrinsic model operator at each point. It proves that the Rockland condition at every point characterizes hypoellipticity and the existence of a parametrix, and shows that if the Levi form has constant rank, the inverse depends smoothly on the base point. For sublaplacians, it provides an intrinsic criterion involving the Levi form and the endomorphism $\mu(a)$ that determines invertibility of the principal symbol (and hence hypoellipticity), with a specialized condition $X(k)$ for the horizontal sublaplacian. The framework unifies major hypoellipticity results—Hörmander's sum of squares, the Kohn Laplacian, the horizontal sublaplacian, and the contact Laplacian—demonstrating invariance under Heisenberg diffeomorphisms and compatibility with transposes, and paving the way for intrinsic, global index-theoretic applications in the Heisenberg setting.

Abstract

In this paper we define an intrinsic notion of principal for the Hypoelliptic calculus on Heisenberg manifolds. More precisely, the principal symbol of a \psivdo appears as a homogeneous section over the linear dual of the tangent Lie algebra bundle of the manifold. This definition is an important step towards using global $K$-theoretic tools in the Heisenberg setting, such as those involved in the elliptic setting for proving the Atiyah-Singer index theorem or the regularity of the eta invariant. On the other hand, the intrinsic definition of the principal symbol enables us to give an intrinsic sense to the model operator of \psivdo at point and to give a definitive proof that the Heisenberg calculus is modelled at each point by the calculus of left-invariant \psidos on the tangent group at the point. This also allows us to define an intrinsic Rockland condition for \psivdos which is shown to be equivalent to the invertibility of the principal symbol, provided that the Levi form has constant rank. Furthermore, we review the main hypoellipticity results on Heisenberg manifolds in terms of the results of the paper. In particular, we complete the treatment of the Kohn Laplacian by Beals-Greiner and establish that for the horizontal sublaplacian the invertibility of the principal symbol is equivalent to some condition on the Levi form, called condition $X(k)$. Incidentally, this paper provides us with a relatively up-to-date overview of the main facts about the Heisenberg calculus.

Intrinsic notion of principal symbol for the Heisenberg calculus

TL;DR

This paper defines an intrinsic principal symbol for the hypoelliptic Heisenberg calculus on Heisenberg manifolds, realized as a homogeneous section of over the tangent Lie group bundle , linking tangent-group data with Rockland criteria and parametrix existence while clarifying the intrinsic model operator at each point. It proves that the Rockland condition at every point characterizes hypoellipticity and the existence of a parametrix, and shows that if the Levi form has constant rank, the inverse depends smoothly on the base point. For sublaplacians, it provides an intrinsic criterion involving the Levi form and the endomorphism that determines invertibility of the principal symbol (and hence hypoellipticity), with a specialized condition for the horizontal sublaplacian. The framework unifies major hypoellipticity results—Hörmander's sum of squares, the Kohn Laplacian, the horizontal sublaplacian, and the contact Laplacian—demonstrating invariance under Heisenberg diffeomorphisms and compatibility with transposes, and paving the way for intrinsic, global index-theoretic applications in the Heisenberg setting.

Abstract

In this paper we define an intrinsic notion of principal for the Hypoelliptic calculus on Heisenberg manifolds. More precisely, the principal symbol of a \psivdo appears as a homogeneous section over the linear dual of the tangent Lie algebra bundle of the manifold. This definition is an important step towards using global -theoretic tools in the Heisenberg setting, such as those involved in the elliptic setting for proving the Atiyah-Singer index theorem or the regularity of the eta invariant. On the other hand, the intrinsic definition of the principal symbol enables us to give an intrinsic sense to the model operator of \psivdo at point and to give a definitive proof that the Heisenberg calculus is modelled at each point by the calculus of left-invariant \psidos on the tangent group at the point. This also allows us to define an intrinsic Rockland condition for \psivdos which is shown to be equivalent to the invertibility of the principal symbol, provided that the Levi form has constant rank. Furthermore, we review the main hypoellipticity results on Heisenberg manifolds in terms of the results of the paper. In particular, we complete the treatment of the Kohn Laplacian by Beals-Greiner and establish that for the horizontal sublaplacian the invertibility of the principal symbol is equivalent to some condition on the Levi form, called condition . Incidentally, this paper provides us with a relatively up-to-date overview of the main facts about the Heisenberg calculus.

Paper Structure

This paper contains 29 sections, 48 theorems, 209 equations.

Table of Contents

  1. Introduction
  2. Intrinsic notion of principal symbol for the Heisenberg calculus
  3. Rockland condition and hypoellipticity
  4. Hypoellipticity criterions for sublaplacians
  5. Main examples of hypoelliptic operators on Heisenberg manifolds
  6. Organization of the paper
  7. Heisenberg manifolds and their tangent Lie group bundles
  8. The tangent Lie group bundle
  9. Heisenberg coordinates and nilpotent approximation of vector fields
  10. Hypoelliptic calculus on Heisenberg manifolds
  11. Left-invariant pseudodifferential operators
  12. $\mathbf{\Psi_{H}}$DO's on an open subset of $\mathbb{R}^{d+1}$
  13. Composition of $\mathbf{\Psi_{H}}$DO's
  14. The distribution kernels of $\mathbf{\Psi_{H}}$DO's
  15. $\mathbf{\Psi_{H}}$DO's on a general Heisenberg manifold
  16. ...and 14 more sections

Key Result

Theorem 1.1

Let P:C^{\infty}_{c}(M,\mathcal{E})\rightarrow C^{\infty}(M,\mathcal{E}) be a \Psi_{H}DO of order m. Then there exists a unique symbol a unique symbol \sigma_{m}(P)\in S_{m}(\mathfrak{g}^{*}M, \mathcal{E}) such that, for any a \in M, the symbol \sigma_{m}(P)(a,.) agrees in trivializing Heisenberg co

Theorems & Definitions (95)

  • Theorem 1.1
  • Proposition 1.2
  • Proposition 1.3
  • Proposition 1.4
  • Proposition 1.5
  • Proposition 1.6
  • Proposition 1.7
  • Proposition 1.8
  • Proposition 1.9
  • Definition 2.1
  • ...and 85 more