[Paper Review] Functional calculus and spectral asymptotics for hypoelliptic operators on Heisenberg Manifolds. I

Raphaël Ponge|ArXiv.org|Feb 25, 2005

Spectral Theory in Mathematical Physics49 references3 citations

TL;DR

This paper develops a functional calculus for hypoelliptic operators on Heisenberg manifolds using the Heisenberg calculus of Beals-Greiner and Taylor, establishing complex powers as holomorphic families of $Ψ_{H}$ DOs and deriving sharp spectral asymptotics. It provides criteria for inverting the heat operator $P + \partial_t$ and proves Weyl asymptotics for geometric operators like the Kohn Laplacian and contact Laplacian, with explicit constants depending only on dimension and form degree.

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.

Motivation & Objective

• To develop a functional calculus for hypoelliptic operators on Heisenberg manifolds using the Heisenberg calculus, overcoming limitations of classical pseudodifferential methods.
• To establish complex powers of hypoelliptic operators as holomorphic families of $Ψ_{H}$ DOs, enabling sharp regularity estimates via a scale of weighted Sobolev spaces.
• To derive criteria on the principal symbol for inverting the heat operator $P + \partial_t$ and obtaining small-time heat kernel asymptotics.
• To prove Weyl asymptotics for key geometric operators on CR and contact manifolds, including the Kohn Laplacian, horizontal sublaplacian, and contact Laplacian.
• To lay the foundation for a noncommutative geometric framework in the Heisenberg setting, including Connes-Moscovici's local index formula.

Proposed method

• Utilizes the Heisenberg calculus of Beals-Greiner and Taylor, which models operators on the nilpotent tangent group $G_xM$ at each point $x \in M$.
• Represents the heat kernel via a pseudodifferential approach, avoiding the standard Seeley method due to lack of microlocality in the Heisenberg calculus.
• Constructs complex powers of hypoelliptic operators as holomorphic families of $Ψ_{H}$ DOs, enabling interpolation between Folland-Stein weighted Sobolev spaces.
• Applies the tangent groupoid construction from [Po6] to define principal symbols intrinsically, enabling symbolic calculus on general Heisenberg manifolds.
• Employs Heisenberg coordinates centered at a point $a \in M$ to compare local model operators on $\mathbb{H}^{2n+1}$ with global operators on $M$.
• Uses the model operator $\Delta_R^0$ on the Heisenberg group $\mathbb{H}^{2n+1}$ to derive asymptotics via the fundamental solution $K_0(x,t)$ of $\Delta_R^0 + \partial_t$.

Experimental results

Research questions

• RQ1 How can complex powers of hypoelliptic operators be defined in the Heisenberg calculus to yield sharp regularity estimates?
• RQ2 What conditions on the principal symbol of a hypoelliptic operator $P$ ensure invertibility of the heat operator $P + \partial_t$?
• RQ3 What is the small-time heat kernel asymptotics for hypoelliptic operators on Heisenberg manifolds?
• RQ4 How do Weyl asymptotics for the Kohn Laplacian and contact Laplacian manifest geometrically on CR and contact manifolds?
• RQ5 Can the spectral theory of hypoelliptic operators be used to construct a noncommutative geometric framework in the Heisenberg setting?

Key findings

• For $k=0,\ldots,2n$, $k \neq n$, the eigenvalue counting function $N(\Delta_{R|_{\Lambda^k}})$ satisfies $N(\Delta_{R|_{\Lambda^k}}) \sim \nu_{nk}(\operatorname{vol}_\theta M)\lambda^{n+1}$ as $\lambda \to \infty$, with $\nu_{nk} = 2^{-n}/(n+1)! \cdot \operatorname{tr}_{\Lambda^{k,0}} K_0|_{\Lambda^{k,0}}(0,1)$.
• For $j=1,2$, the eigenvalue counting function $N(\Delta_{R|_{\Lambda^n_j}})$ satisfies $N(\Delta_{R|_{\Lambda^n_j}}) \sim \nu_n^{(j)}(\operatorname{vol}_\theta M)\lambda^{(n+1)/2}$ as $\lambda \to \infty$, with $\nu_n^{(j)} = 2^{-n}\Gamma(1 + (n+1)/2)^{-1} \cdot \operatorname{tr}_{\Lambda^{n,0}_j} K_0|_{\Lambda^{n,0}_j}(0,1)$.
• The constants $\nu_{nk}$ and $\nu_n^{(j)}$ depend only on $n$ and $k$ (or $j$), making them universal constants independent of the specific manifold.
• The model operator $\Delta_R^a$ at a point $a$ agrees with the contact Laplacian $\Delta_R^0$ on $\mathbb{H}^{2n+1}$ in Heisenberg coordinates, enabling local asymptotic analysis.
• The heat kernel asymptotics are derived via the fundamental solution $K_0(x,t)$ of $\Delta_R^0 + \partial_t$, with the trace of its restriction to $\Lambda^{k,0}$ and $\Lambda^{n,0}_j$ at $t=1$ determining the spectral constants.
• The Weyl asymptotics are geometrically reformulated: the eigenvalue growth rates reflect the intrinsic dimension and structure of the contact manifold, with distinct exponents for $k \neq n$ and $k=n$.