First, let us explain the precise meaning of the compressed title. The word “eigenvalues” means the first nontrivial Neumann or Dirichlet eigenvalues, or the principal eigenvalues. The word “inequalities” means the Poincar ́e inequalities, the logarithmic Sobolev inequalities, the Nash inequalities, and so on. Actually, the first eigenvalues can be described by some Poincar ́e inequali- ties, and so the second topic has a wider range than the first one. Next, for a Markov process, corresponding to its operator, each inequality describes a type of ergodicity. Thus, study of the inequalities and their relations provides a way to develop the ergodic theory for Markov processes. Due to these facts, from a probabilistic point of view, the book can also be regarded as a study of “ergodic convergence rates of Markov processes,” which could serve as an alternative title of the book. However, this book is aimed at a larger class of readers, not only probabilists.
The importance of these topics should be obvious. On the one hand, the first eigenvalue is the leading term in the spectrum, which plays an important role in almost every branch of mathematics. On the other hand, the ergodic convergence rates constitute a recent research area in the theory of Markov processes. This study has a very wide range of applications. In particular, it provides a tool to describe the phase transitions and the effectiveness of random algorithms, which are now a very fashionable research area.