Communications in Mathematical Sciences

Volume 5 (2007)

Number 4

Interpolation between logarithmic Sobolev and Poincare inequalities

Pages: 971 – 979



Anton Arnold

Jean-Philippe Bartier

Jean Dolbeault


This paper is concerned with intermediate inequalities which interpolate between the logarithmic Sobolev (LSI) and the Poincaré inequalities. Assuming that a given probability measure gives rise to a LSI, we derive generalized Poincaré inequalities, improving upon the known constants from the literature. We also analyze the special case when these inequalities are restricted to functions with zero components for the first eigenspaces of the corresponding evolution operator.


functional inequalities; Poincaré inequality; logarithmic Sobolev inequality; spectral gap; hypercontractivity

2010 Mathematics Subject Classification

35K10, 39B62, 46E35, 60F10, 60J60

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Published 1 January 2007