[Paper Review] Pseudodifferential calculus on a singular foliation

Iakovos Androulidakis, Georges Skandalis|ArXiv.org|Sep 7, 2009

Advanced Operator Algebra Research4 references3 citations

TL;DR

This paper establishes a pseudodifferential calculus on singular foliations using their holonomy groupoid and associated C*-algebra. It constructs a Laplacian operator as a positive, unbounded, self-adjoint operator on $L^2(M)$, proving its regularity and ellipticity via the calculus, which extends to a full index theory framework on compact manifolds.

ABSTRACT

In a previous paper ([1]), we associated a holonomy groupoid and a C*-algebra to any singular foliation (M,F). Using these, we construct the associated pseudodifferential calculus. This calculus gives meaning to a Laplace operator of any singular foliation F on a compact manifold M, and we show that it can be naturally understood as a positive, unbounded, self-adjoint operator on L2(M).

Motivation & Objective

• To extend pseudodifferential calculus to singular foliations, which lack the regularity of regular foliations.
• To define a longitudinal Laplacian operator on a compact manifold equipped with a singular foliation.
• To show that this Laplacian is a positive, unbounded, self-adjoint operator on $L^2(M)$, using the groupoid and $C^*$-algebra framework.
• To establish a filtration of Sobolev spaces and a $C^*$-algebraic structure for pseudodifferential operators of negative and zero order.
• To lay the foundation for a full index theory on singular foliations through the construction of a regular, elliptic, positive-order operator.

Proposed method

• Constructs a pseudodifferential calculus using bi-submersions and oscillatory integrals with symbols on the normal bundle of a bisection.
• Defines kernels via oscillatory integrals: $k_a(u) = ext{int}_{N^*_{p(u)}} a(p(u), heta) e^{i heta ullet ext{exp}^{-1}(u)} ilde{ ho}(u) d heta$, where $\tilde{\rho}$ is a cutoff.
• Uses the principal symbol of a pseudodifferential operator as a homogeneous function on the cosphere bundle $\mathcal{F}^*$, which is a family of vector spaces of non-constant dimension.
• Shows that pseudodifferential kernels define multipliers of the $C^*$-algebra $C^*(M,\mathcal{F})$, and that the algebra of such operators is filtered by order.
• Establishes an exact sequence: $0 \to C^*(M,\mathcal{F}) \to \Psi^*(M,\mathcal{F}) \to B \to 0$, where $B$ is a quotient of $C_0(S^*\mathcal{F})$.
• Applies the calculus to define the Laplacian $\Delta = \sum X_k^*X_k$ from a finite generating set of vector fields in $\mathcal{F}$, proving it is longitudinally elliptic and regular.

Experimental results

Research questions

• RQ1 How can a pseudodifferential calculus be defined on a singular foliation, given the lack of regularity in the holonomy groupoid?
• RQ2 Can a Laplacian operator be meaningfully defined on a singular foliation and shown to be self-adjoint on $L^2(M)$?
• RQ3 What is the role of the $C^*$-algebra of the foliation in realizing pseudodifferential operators as multipliers?
• RQ4 How does the longitudinal ellipticity of an operator relate to its regularity and invertibility in the $C^*$-algebraic framework?
• RQ5 Can the Sobolev space filtration and the $\Psi^{-\infty}$-calculus be extended to singular foliations to support index theory?

Key findings

• The Laplacian $\Delta = \sum X_k^*X_k$ is a longitudinally elliptic operator of order 2, and thus defines a regular, unbounded, self-adjoint multiplier of $C^*(M,\mathcal{F})$.
• The Laplacian induces a self-adjoint operator on $L^2(M)$, with the same spectrum under weakly equivalent representations, including the natural $L^2(M)$-representation.
• Negative-order pseudodifferential operators are elements of the full and reduced $C^*$-algebra of the foliation.
• Zero-order pseudodifferential operators define bounded multipliers of the $C^*$-algebra, and the algebra $\Psi^*(M,\mathcal{F})$ fits into an exact sequence with $C^*(M,\mathcal{F})$ and $C_0(S^*\mathcal{F})$.
• The calculus is closed under composition and pullback, and the principal symbol depends only on the germ of the symbol on $\mathcal{F}^*$.
• The algebra of smoothing operators $\Psi^{-\infty}(\mathcal{U})$ contains $\mathcal{A}_{\mathcal{U}}$ densely, and the Sobolev spaces $H^k(P)$ are well-defined for longitudinally elliptic operators $P$.