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Summary: INTERPOLATION BETWEEN LOGARITHMIC SOBOLEV AND
POINCARŽE INEQUALITIES
ANTON ARNOLD, JEAN-PHILIPPE BARTIER, AND JEAN DOLBEAULT
Abstract. This paper is concerned with intermediate inequalities which interpolate between
the logarithmic Sobolev (LSI) and the PoincarŽe inequalities. Assuming that a given probability
measure gives rise to a LSI, we derive generalized PoincarŽe 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 on the first eigenspaces of the corresponding evolution operator.
1. Introduction
In 1989 W. Beckner [B] derived a family of generalized PoincarŽe inequalities
(GPI) for the Gaussian measure that yield a sharp interpolation between the classical
PoincarŽe inequality and the logarithmic Sobolev inequality (LSI) of L. Gross [G]. For
any 1 p < 2 these GPIs read
1
2 - p Rd
f2
d”0 -
Rd
|f|p
d”0
|
