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Functional calculus and spectral asymptotics for hypoelliptic operators on Heisenberg Manifolds. I

Raphael Ponge

TL;DR

The paper develops a robust framework for hypoelliptic analysis on Heisenberg manifolds by extending Beals-Greiner and Taylor’s Heisenberg calculus to holomorphic families of Ψ_H DOs and to complex powers. It introduces weighted Heisenberg Sobolev spaces, establishes invertibility criteria for heat operators via Rockland-type conditions, and derives general Weyl and heat-kernel asymptotics, with geometric formulations for CR and contact structures. The approach hinges on heat-kernel representations and an almost homogeneous symbol/kernel calculus to handle non-elliptic, hypoelliptic settings where Seeley’s method fails. These tools yield precise spectral data for key geometric operators (Kohn Laplacian, horizontal sublaplacian, and contact Laplacian) and set the stage for noncommutative-geometric interpretations and further spectral invariants in future work.

Abstract

This paper is part of a series papers devoted to geometric and spectral theoretic applications of the hypoelliptic calculus on Heisenberg manifolds. More specifically, in this paper we make use of the Heisenberg calculus of Beals-Greiner and Taylor to analyze the spectral theory of hypoelliptic operators on Heisenberg manifolds. The main results of this paper include: (i) Obtaining complex powers of hypoelliptic operators as holomorphic families of Psi_{H}DO's, which can be used to define a scale of weighted Sobolev spaces interpolating the weighted Sobolev spaces of Folland-Stein and providing us with sharp regularity estimates for hypoelliptic operators on Heisenberg manifolds; (ii) Criterions on the principal symbol of $P$ to invert the heat operator $P+\partial_{t}$ and to derive the small time heat kernel asymptotics for $P$; (iii) Weyl asymptotics for hypoelliptic operators which can be reformulated geometrically for the main geometric operators on CR and contact manifolds, that is, the Kohn Laplacian, the horizontal sublaplacian and its conformal powers, as well as the contact Laplacian. For dealing with complex powers of hypoelliptic operators we cannot make use of the standard approach of Seeley, so we rely on a new approach based on the pseudodifferential approach representation of the heat kernel. This is especially suitable for dealing with positive hypoelliptic operators. We will deal with more general operator in a forthcoming paper using another new approach.

Functional calculus and spectral asymptotics for hypoelliptic operators on Heisenberg Manifolds. I

TL;DR

The paper develops a robust framework for hypoelliptic analysis on Heisenberg manifolds by extending Beals-Greiner and Taylor’s Heisenberg calculus to holomorphic families of Ψ_H DOs and to complex powers. It introduces weighted Heisenberg Sobolev spaces, establishes invertibility criteria for heat operators via Rockland-type conditions, and derives general Weyl and heat-kernel asymptotics, with geometric formulations for CR and contact structures. The approach hinges on heat-kernel representations and an almost homogeneous symbol/kernel calculus to handle non-elliptic, hypoelliptic settings where Seeley’s method fails. These tools yield precise spectral data for key geometric operators (Kohn Laplacian, horizontal sublaplacian, and contact Laplacian) and set the stage for noncommutative-geometric interpretations and further spectral invariants in future work.

Abstract

This paper is part of a series papers devoted to geometric and spectral theoretic applications of the hypoelliptic calculus on Heisenberg manifolds. More specifically, in this paper we make use of the Heisenberg calculus of Beals-Greiner and Taylor to analyze the spectral theory of hypoelliptic operators on Heisenberg manifolds. The main results of this paper include: (i) Obtaining complex powers of hypoelliptic operators as holomorphic families of Psi_{H}DO's, which can be used to define a scale of weighted Sobolev spaces interpolating the weighted Sobolev spaces of Folland-Stein and providing us with sharp regularity estimates for hypoelliptic operators on Heisenberg manifolds; (ii) Criterions on the principal symbol of to invert the heat operator and to derive the small time heat kernel asymptotics for ; (iii) Weyl asymptotics for hypoelliptic operators which can be reformulated geometrically for the main geometric operators on CR and contact manifolds, that is, the Kohn Laplacian, the horizontal sublaplacian and its conformal powers, as well as the contact Laplacian. For dealing with complex powers of hypoelliptic operators we cannot make use of the standard approach of Seeley, so we rely on a new approach based on the pseudodifferential approach representation of the heat kernel. This is especially suitable for dealing with positive hypoelliptic operators. We will deal with more general operator in a forthcoming paper using another new approach.

Paper Structure

This paper contains 43 sections, 103 theorems, 353 equations.

Table of Contents

  1. Introduction
  2. Holomorphic families of $\Psi_{H}$DO's
  3. Complex powers of hypoelliptic operators
  4. Weighted Sobolev spaces
  5. Rockland condition and the heat equation
  6. Spectral asymptotics for hypoelliptic operators
  7. Weyl asymptotics and CR geometry
  8. Weyl asymptotics and contact geometry
  9. Organization of the paper
  10. Heisenberg manifolds and their tangent Lie group bundles
  11. The tangent Lie group bundle
  12. Heisenberg coordinates and nilpotent approximation of vector fields
  13. Most common operators on Heisenberg manifolds
  14. Hörmander's sum of squares.
  15. The Kohn Laplacian on a CR manifold
  16. ...and 28 more sections

Key Result

Proposition 1.1

For j=1,2 let (P_{j,z})_{z \in \Omega}\subset \Psi_{H}^{*}(M,\mathcal{E}) be a holomorphic family of \Psi_{H}DO's and supposed that (P_{1,z})_{z \in \Omega} or (P_{2,z})_{z \in \Omega} is uniformly properly supported with respect to z. Then the family of products (P_{1,z}P_{2,z})_{z \in \Omega} is a

Theorems & Definitions (211)

  • Proposition 1.1
  • Proposition 1.2
  • Proposition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Proposition 1.7
  • Proposition 1.8
  • Proposition 1.9
  • Proposition 1.10
  • ...and 201 more