Heisenberg calculus and spectral theory of hypoelliptic operators on Heisenberg manifolds

Ponge, Raphaël

doi:10.1090/memo/0906

Cited by 55 publications

(131 citation statements)

References 73 publications

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“…The setup and results can be easily generalised to pseudodifferential operators and more general manifolds. One could apply the same approach to a hypoelliptic operator A on a Heisenberg manifold [25] with heat kernel expansion as in [4]. The heat kernel of f (A) would have the probabilistic interpretation of the transition density for a subordinate process.…”

Section: Corollary 37 Under the Assumptions Of Theorem 35 We Have mentioning

confidence: 99%

Off-Diagonal Heat Kernel Asymptotics of Pseudodifferential Operators on Closed Manifolds and Subordinate Brownian Motion

Fahrenwaldt

2017

Integr. Equ. Oper. Theory

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Abstract.We derive the off-diagonal short-time asymptotics of the heat kernels of functions of generalised Laplacians on a closed manifold. As an intermediate step we give an explicit asymptotic series for the kernels of the complex powers of generalised Laplacians. Each asymptotic series is formulated in terms of the geodesic distance. The key application concerns upper bounds for the transition density of subordinate Brownian motion. The approach is highly explicit and tractable.Mathematics Subject Classification. Primary 35K08; Secondary 58J65, 35S05.

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Section: Corollary 37 Under the Assumptions Of Theorem 35 We Have mentioning

confidence: 99%

Off-Diagonal Heat Kernel Asymptotics of Pseudodifferential Operators on Closed Manifolds and Subordinate Brownian Motion

Fahrenwaldt

2017

Integr. Equ. Oper. Theory

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“…. , W m a basis of the first layer (the horizontal layer) of the Lie algebra g of G. The following equivalence is known ([9, Theorem 2.1 and following remark], and also [16,Proposition 1.4.7], for vector bundles over Heisenberg groups). (i) L is hypoelliptic.…”

Section: Introductionmentioning

confidence: 99%

Hypoellipticity, fundamental solution and Liouville type theorem for matrix-valued differential operators in Carnot groups

Baldi¹,

Franchi²,

Tesi³

2009

J. Eur. Math. Soc.

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Abstract. Let L be a non-negative self-adjoint N × N matrix-valued operator of order a ≤ Q on a Carnot group G. Here Q is the homogeneous dimension of G. The aim of this paper is to investigate the relationship between hypoellipticity and maximal hypoellipticity (i.e. sharp L 2 estimates in appropriate Sobolev spaces), L p -maximal hypoellipticity (i.e. sharp L p estimates in appropriate Sobolev spaces for 1 < p < ∞), and what we call maximal subellipticity of L (which is basically a sharp higher order energy estimate).

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“…The Heisenberg calculus was built independently by Beals, Greiner [1] and Taylor [63] as the relevant pseudodifferential tool to study the main geometric operators on contact and CR manifolds, which fail to be elliptic, but may be hypoelliptic (see also [7,18,23,51]). This calculus holds in the general setting of a Heisenberg manifold, that is, a manifold M together with a distinguished hyperplane bundle H ⊂ TM, and we construct a noncommutative residue trace in this general context.…”

Section: Introductionmentioning

confidence: 99%

Noncommutative residue for Heisenberg manifolds. Applications in CR and contact geometry

Ponge

2007

Journal of Functional Analysis

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This paper has four main parts. In the first part, we construct a noncommutative residue for the hypoelliptic calculus on Heisenberg manifolds, that is, for the class of Ψ H DO operators introduced by Beals-Greiner and Taylor. This noncommutative residue appears as the residual trace on integer order Ψ H DOs induced by the analytic extension of the usual trace to non-integer order Ψ H DOs. Moreover, it agrees with the integral of the density defined by the logarithmic singularity of the Schwartz kernel of the corresponding Ψ H DO. In addition, we show that this noncommutative residue provides us with the unique trace up to constant multiple on the algebra of integer order Ψ H DOs. In the second part, we give some analytic applications of this construction concerning zeta functions of hypoelliptic operators, logarithmic metric estimates for Green kernels of hypoelliptic operators, and the extension of the Dixmier trace to the whole algebra of integer order Ψ H DOs. In the third part, we present examples of computations of noncommutative residues of some powers of the horizontal sublaplacian and the contact Laplacian on contact manifolds. In the fourth part, we present two applications in CR geometry. First, we give some examples of geometric computations of noncommutative residues of some powers of the horizontal sublaplacian and of the Kohn Laplacian. Second, we make use of the framework of noncommutative geometry and of our noncommutative residue to define lower-dimensional volumes in pseudohermitian geometry, e.g., we can give sense to the area of any 3-dimensional CR manifold endowed with a pseudohermitian structure. On the way we obtain a spectral interpretation of the Einstein-Hilbert action in pseudohermitian geometry.

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Heisenberg calculus and spectral theory of hypoelliptic operators on Heisenberg manifolds

Cited by 55 publications

References 73 publications

Off-Diagonal Heat Kernel Asymptotics of Pseudodifferential Operators on Closed Manifolds and Subordinate Brownian Motion

Off-Diagonal Heat Kernel Asymptotics of Pseudodifferential Operators on Closed Manifolds and Subordinate Brownian Motion

Hypoellipticity, fundamental solution and Liouville type theorem for matrix-valued differential operators in Carnot groups

Noncommutative residue for Heisenberg manifolds. Applications in CR and contact geometry

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