Title Page
Abstract
Contents
Chapter 1. Introduction 9
Chapter 2. Harnack inequality for Nonlocal operators on Manifolds with nonnegative curvature 14
2.1. Introduction 14
2.1.1. Nonlocal operators on Riemannian manifolds 15
2.1.2. Main results 18
2.1.3. Outline 21
2.2. Preliminaries 22
2.3. Discrete ABP-type estimates 28
2.4. A barrier function 42
2.5. Lε-estimate[이미지참조] 47
2.6. Harnack inequality 50
2.7. Hölder estimates 55
Chapter 3. Harnack inequality for fractional Laplacian-type operators on hyperbolic spaces 59
3.1. Introduction 59
3.2. Preliminaries 66
3.2.1. Hyperbolic spaces 66
3.2.2. Fractional Laplacian on the hyperbolic spaces 68
3.2.3. Hyperbolic spaces revisited 71
3.3. Scale functions 72
3.4. Discrete ABP-type estimates 77
3.5. A barrier function 88
3.6. Lε-estimate[이미지참조] 95
3.7. Harnack inequality 98
3.8. Hölder estimates 102
3.9. Appendix 105
3.9.1. Special functions 105
Chapter 4. The fractional p-Laplacian on hyperbolic spaces 112
4.1. Introduction 112
4.2. Preliminaries 117
4.2.1. The hyperbolic space 117
4.2.2. The modified Bessel function 118
4.3. Pointwise representation with singular kernel 119
4.4. Extension problem 124
4.5. Pointwise convergence 129
4.6. Appendix 146
4.6.1. Auxiliary result 146
Bibliography 147
국문초록 159