Laplacians on Smooth Distributions as C * -Algebra Multipliers
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LAPLACIANS ON SMOOTH DISTRIBUTIONS AS C ∗ -ALGEBRA MULTIPLIERS UDC 514.7, 517.9
Yu. A. Kordyukov
Abstract. In this paper, we continue the study of spectral properties of Laplacians associated with an arbitrary smooth distribution on a compact manifold started in a previous paper. Under the assumption that the singular foliation generated by the distribution is smooth, we prove that the Laplacian associated with the distribution defines an unbounded, regular, self-adjoint operator in some Hilbert module over the C ∗ -algebra of the foliation. Keywords and phrases: foliation, Hilbert module, Laplacian, hypoelliptic operator, smooth distribution, multiplier. AMS Subject Classification: 58J60, 53C17, 46L08, 58B34
CONTENTS 1. Introduction . . . . . . . . . . . . . . . 2. Operators in Hilbert Modules . . . . . 3. Families of Hypoelliptic Operators and References . . . . . . . . . . . . . . . .
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Introduction
In this paper, we continue the study of spectral properties of Laplacians associated with an arbitrary smooth distribution on a compact manifold initiated in [17]. First, we recall their definition. Let M be a connected, compact, smooth manifold of dimension n equipped with a smooth positive density μ. Let H be a smooth distribution on M of rank d (that is, H is a smooth subbundle of the tangent bundle T M of M ) and g be a smooth fiberwise inner product on H. We define the horizontal differential dH f of a function f ∈ C ∞ (M ) as the restriction of its differential df to H ⊂ T M . Thus, dH f is a section of the dual bundle H ∗ of H, dH f ∈ C ∞ (M, H ∗ ), and we get a well-defined firstorder differential operator dH : C ∞ (M ) → C ∞ (M, H ∗ ). The Riemannian metric g and the positive smooth density μ induce inner products on C ∞ (M ) and C ∞ (M, H ∗ ), which allow us to consider the adjoint operator d∗H : C ∞ (M, H ∗ ) → C ∞ (M ) of dH . The Laplacian ΔH associated with (H, g, μ) is a second-order differential operator on C ∞ (M ) given by ΔH = d∗H dH . Let X1 , . . . , Xd be a local orthonormal frame in H defined on an open subset Ω ⊂ M . It can be easily verified that the restriction of the operator ΔH to Ω is given by d Xj∗ Xj . ΔH = Ω
j=1
Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 152, Mathematical Physics, 2018.
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c 2021 Springer Science+Business Media, LLC 1072–3374/21/2522–0190
The operator ΔH can be also defined by means of the associated quadratic form (ΔH u, u) = |dH u(x)|2g dμ(x), u ∈ C ∞ (M ). M
We will consider the Laplacian ΔH as an unbounded linear operator in the Hilbert space L2 (M, μ) with initial domain C ∞ (M ). Theorem 1.1 (see [17]). The operator ΔH is an essentially self-adjoint operator in L2 (M, μ) (i.e., its closure is a self-adjoint operator ).
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