Abstract
This paper gives a reasonably self-contained account for some recent progress on functional inequalities, semigroup properties and spectrum estimates. Two sorts of functional inequalities are considered, they are actually equivalent and are general forms of Sobolev type inequalities. Semigroup properties, 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 diffusions on Riemannian manifolds and jump processes. Most yet unpublished results are reproved.
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