Functional Inequalities, Markov Semigroups and Spectral Theory - Mathematics Monograph Series 4

In this book, we introduce functional inequalities to describe:
(i) the spectrum of the generator: the essential and discrete spectrums, high order eigenvalues, the principal eigenvalue, and the spectral gap;
(ii) the semigroup properties: the uniform integrability, the compactness, the convergence rate, and the existence of density;
(iii) the reference measure and the intrinsic metric: the concentration, the isoperimetric inequality, and the transportation cost inequality.

For reader’s convenience and for the completeness of the account, we summarize some necessary preliminaries in Chapter 0. Corresponding to various levels of spectral and semigroup properties, chapter 1,3,4,5 and 6 focus on several different functional inequalities respectively. And Chapter 2 is devoted to diffusion processes on Riemannian manifolds and applications to geometry analysis.

Chapter 0 Preliminaries
Chapter 1 Poincare Inequality and Spectral Gap
Chapter 2 Diffusion Processes on Manifolds and Applications
Chapter 3 Functional Inequalities and Essential Spectrum
Chapter 4 Weak Poincare Inequalities and Convergence of Semigroups
Chapter 5 Log-Sobolev Inequalities and Semigroup Properties
Chapter 6 Interpolations of Poincare and Log-Sobolev Inequalities
Chapter 7 Some Infinite Dimensional Models