Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces.

Intro -- Contents -- Preface -- Some questions on elliptic operators -- Introduction -- 1. Operators of order two -- 2. Higher order operators and systems -- 3. Bounded elliptic forms -- References -- Heat kernels and sets with fractal structure -- 1. Fractal sets -- 2. The graphical Sierpinski gasket -- 2.1 Graphs and random walks -- 2.2 Simple random walk on the graphical SG -- 2.3 Analysis on the Sierpinski gasket -- 3. Sierpinski carpets -- 3.1 Basic properties -- 3.2 Resistance and crossing times -- 4. General results on pre-fractal graphs -- 4.1 On-diagonal upper bounds -- 4.2 Off-diagonal upper bounds -- 4.3 Lower bounds -- 4.4 Stability results for general pre-fractal graphs -- 5. Continuum limits -- References -- Brownian motions on compact groups of infinite dimension -- 1. Introduction -- 2. Splitting into abelian and semisimple parts -- 3. The abelian and semisimple cases -- 4. Potential theory -- 5. Examples -- References -- Heat kernel and isoperimetry on non-compact Riemannian manifolds -- 1. Introduction -- 2. A general scale of Sobolev inequalities -- 3. L2 isoperimetric profile and heat kernel decay -- 4. How to compute isoperimetric profiles -- References -- Heat kernels measures and infinite dimensional analysis -- 1. Introduction -- 2. Finite Dimensional Heat Kernel Measures -- 3. Describing Smooth Measures on Rd Without Reference to Lebesgue Measure -- 4. Infinite Dimensional Considerations -- 5. Heat Kernel Measures Associated to l2 -- 6. Classical Wiener Measure -- 7. Path and Loop Group Extensions -- 8. Wiener Measure on W (M) and its Properties -- 9. Motivations -- References -- Heat kernels and function theory on metric measure spaces -- 1. Introduction -- 2. Definition of a heat kernel and examples -- 3. Volume of balls -- 4. Energy form -- 5. Besov spaces and energy domain.

6. Intrinsic characterization of walk dimension -- 7. Inequalities for walk dimension -- 8. Embedding of Besov spaces into Hölder spaces -- 9. Bessel potential spaces -- References -- Sobolev spaces on metric-measure spaces -- 1. Introduction -- 2. Classical Sobolev spaces -- 3. Curves in metric spaces -- 4. Borel and doubling measures -- 5. Modulus of the path family -- 6. Upper gradient -- 7. Sobolev spaces N1,p -- 8. Sobolev spaces M1,p -- 9. Sobolev spaces P1,p -- 10. Abstract derivative and Sobolev spaces H1,p -- 11. Spaces supporting Poincaré inequality -- 12. Historical notes -- References -- Quasiregular mappings and the p-Laplace operator -- 1. Introduction -- 2. A-harmonic functions -- 3. Morphism property and its consequences -- 4. Modulus and capacity inequalities -- 5. Liouville-type results for A-harmonic functions -- References -- Spherical inversion on SL2 (C) -- 1. lwasawa decomposition and characters -- 2. Haar measures -- 3. The Harish transform and the orbital integral -- 4. The Mellin transform S = MH -- 5. Computation of the orbital intgeral -- 6. Gaussians on G -- 7. Polar Haar measure and spherical inversion -- 8. The gaussian-Dirac family -- 9. The heat gaussian -- 10. Appendix. Orbital integrals and the trace formula -- References -- Spectral geometry of crystal lattices -- Lectures on isoperimetric and isocapacitary inequalities in the theory of Sobolev spaces -- Preface -- 1. Introduction -- 2. Classical isoperimetric inequality and its applications to integral inequalities -- 3. Sobolev type inequality for functions with unrestricted boundary values -- 4. Compactness criterion -- 5. The case p = 1, q &lt -- 1 in the Sobolev type inequality (3.1) -- 6. Imbeddings into fractional Sobolev spaces -- 7. Imbedding into a Riesz potential space.

8. Capacity minimizing functions and their applications to Sobolev type inequalities -- 9. An application of p-capacity to Poincaré's inequality -- References -- Some topics related to analysis on metric spaces -- Probability measures on metric spaces of nonpositive curvature -- 1. Geodesic Spaces -- 2. Global NPC Spaces -- 3. Examples of Global NPC Spaces -- 4. Barycenters -- 5. Identification of Barycenters -- 6. Jensen's Inequality and L1 Contraction Property -- 7. Convex Means -- 8. Local NPC Spaces -- References -- Generating function techniques for random walks on graphs -- 1. Introduction -- 2. An incomplete survey of some results -- 3. Generating functions -- 4. First walks on trees -- 5. Free products, n-3/ 2, and a surprising result of Cartwright -- 6. Finite range random walks on free groups -- 7. Uniform space-time estimates on trees -- 8. Sierpiński graphs, and the use of Singularity Analysis -- 9. Final remarks -- References.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.