Functional inequalities, semigroup properties and spectrum estimates

This paper gives a reasonably self-contained account for some recent progre ss on functional inequalities, semigroup properties and spectrum estimates. Two sorts of functional inequalities are considered, they are actually equ ivalent and are general forms of Sobolev type inequalities. Semigroup prope rties, spectrum estimates and concentration of measures are described using these inequalities. Some criteria of functional inequalities and estimates of the spectral gap and the log-Sobolev constant are presented for diffusi ons on Riemannian manifolds and jump processes. Most yet unpublished result s are reproved.