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Differential Operators on Non-compact Harmonic Manifolds

Oliver Brammen

Abstract

We study the algebra of differential operators on non-compact simply connected harmonic manifolds and provide sufficient conditions for them to have a radial fundamental solution and be surjective on the space of smooth function. Furthermore, we show that the algebra of differential operators that commute with taking averages over geodesic spheres is generated by the Laplacian. As an application of this, we show that the heat-semi group is dense in the radial $L^1$ space of a non-compact simply connected harmonic manifold. In the process, we provide a characterisation of the eigenfunctions of the Laplacian and therefore of the eigenfunctions of all differential operators commuting with taking spherical averages.

Differential Operators on Non-compact Harmonic Manifolds

Abstract

We study the algebra of differential operators on non-compact simply connected harmonic manifolds and provide sufficient conditions for them to have a radial fundamental solution and be surjective on the space of smooth function. Furthermore, we show that the algebra of differential operators that commute with taking averages over geodesic spheres is generated by the Laplacian. As an application of this, we show that the heat-semi group is dense in the radial space of a non-compact simply connected harmonic manifold. In the process, we provide a characterisation of the eigenfunctions of the Laplacian and therefore of the eigenfunctions of all differential operators commuting with taking spherical averages.
Paper Structure (10 sections, 30 theorems, 84 equations)

This paper contains 10 sections, 30 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Prilimenaries
  3. The Convulotion on Harmonic manifolds
  4. Convolution Invariant Operators
  5. Charatarisation of Eigenfunctions of The Laplacian
  6. The Abel Transform and Its Dual
  7. Differential Operators
  8. Existance of Fundamental Solutions for Invariant Differential Operators with Constant Horospherical Part
  9. The Algebra of Radial Invariant Operators
  10. The heat kernel

Key Result

Proposition 2.3

A simply connected non-compact harmonic manifold $(X,g)$ is Einstein and therefore the Riemannian metric $g$ and all functions derived from g are analytic. Especially the Laplace operator $\Delta$ is an analytical elliptic operator.

Theorems & Definitions (69)

  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3
  • Definition 2.4
  • Remark 2.5
  • Lemma 2.6: Biswas2019, Lemma 3.1
  • Definition 3.1
  • Remark 3.2
  • Lemma 3.3: PS15,Biswas2019
  • Definition 3.4
  • ...and 59 more