Functional inequalities, semigroup properties and spectrum estimates. (English) Zbl 1037.47505
Summary: This paper gives a reasonably self-contained account of 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 Most as yet unpublished results are reproved.
MSC:
| 47D03 | Groups and semigroups of linear operators |
| 58J60 | Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.) |
| 47-02 | Research exposition (monographs, survey articles) pertaining to operator theory |
| 58J70 | Invariance and symmetry properties for PDEs on manifolds |
| 39B72 | Systems of functional equations and inequalities |
Keywords:
Poincaré inequality; Sobolev type inequalities; concentration of measures; functional inequalities; spectral gap; log-Sobolev constant; diffusions; jump processesReferences:
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