Heat kernel estimates and Lp-spectral theory of locally symmetric spaces

Andreas Weber3866441088, 9783866441088

In this work we derive upper Gaussian bounds for the heat kernel on locally symmetric spaces of non-compact type. Furthermore, we determine explicitly the Lp-spectrum of locally symmetric spaces M whose universal covering is a rank one symmetric space of non-compact type if either the fundamental group of M is small (in a certain sense) or if the fundamental group is arithmetic and M is non-compact.

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Table of contents :
0.1 Heat Diffusion and Heat Equation……Page 15
0.2 Lp-Spectral Theory……Page 16
0.2.1 Volume Growth and Lp-Spectrum……Page 17
0.4 Outline of Chapter 1 — Chapter 6……Page 19
1.1 Algebraic Description of Symmetric Spaces……Page 23
1.3 Locally Symmetric Spaces……Page 26
1.4 Decomposition Theorems……Page 27
2.1 The Heat Kernel on a Riemannian Manifold……Page 31
2.2 The Heat Semigroup on Lp-Spaces……Page 33
2.3 Lp-Spectrum of a Product Manifold……Page 36
2.4 The Heat Kernel on a Symmetric Space……Page 37
2.5 Quotients of Cartan-Hadamard Manifolds……Page 40
3.1 General Compact Riemannian Manifolds……Page 45
3.2 Poincaré Series and the Critical Exponent……Page 47
3.2.1 Estimates of the Poincaré Series……Page 48
3.3 Gaussian Bounds — Part 1……Page 50
3.4 Heat Kernels and L2-Spectrum……Page 54
3.4.1 L2-Spectrum of Locally Symmetric Spaces……Page 56
3.5 Gaussian Bounds — Part 2……Page 57
3.6 Lower Bounds……Page 61
3.7 L2-Eigenfunctions of the Laplace-Beltrami operator……Page 62
4 Locally Symmetric Spaces with Small Fundamental Group……Page 65
4.1.2 The Metric in Horocyclic Coordinates……Page 67
4.1.3 Geodesic Compactification and Limit sets……Page 69
4.2 Lp-Spectrum……Page 75
5.1.1 Arithmethic Groups, Q-Rank and R-Rank……Page 83
5.1.2 Siegel Sets and Reduction Theory……Page 86
5.1.3 Rational Horocyclic Coordinates……Page 89
5.2 Lp-Spectrum……Page 91
6.1 Definition……Page 97
6.2 Lp-Spectrum and Geometry……Page 98
A Tensor Products……Page 101
Bibliography……Page 103