- LOGARITHMIC SOBOLEV INEQUALITY 247
COROLLARY 6.38 (Log Sobolev inequality, version 2). For any b > 0,
there exists a constant C (b, g) such that if a function ¢ satisfies JM e-<P dμ =
1, then(6.68)4.2. Logarithmic Sobolev inequality on Euclidean space. We
give a proof of Grass's logarithmic Sobolev inequality on Euclidean space
[170]. Although this result will not be used elsewhere in Part I of this
volume, we include it here since it is both fundamental and elegant.THEOREM 6.39. For any nonnegative function¢ E W^1 '^2 (IRn), we haveNote that the above inequality is scale-invariant, that is, the inequal-
ity is preserved under multiplication of ¢ by a positive constant. Also, if
f JRn ¢^2 dx = 1, then the inequality says that f JRn ¢^2 log ¢ dx ::; f JRn I \7 ¢ J 2 dx.
The following consequence of Grass's logarithmic Sobolev inequality is actu-
ally equivalent to it. (We leave the proof of the equivalence to the reader.)
COROLLARY 6.40. If fJRn (47rr)-n/^2 e-f dx = 1, then
(6.69) Ln (r 1Vfl^2 + f - n) (47rr)-n/^2 e-f dx ~ 0.
In particular, taking T = 1/2, we have
(6.70)provided fJRn (27r)-n/^2 e-f dx = 1. Moreover, if we can perform an integra-
tion by parts, then we may rewrite (6.69) as(6.71) Ln (r (21:)..f-J'VfJ^2 ) + f-n) (47rT)-n/^2 e-f dx ~ 0.
REMARK 6.41. Compare the LHS of (6.69) with the entropy (6.1) and
compare the integrand on the LHS of (6.71) with Perelman's differential
Harnack quantity (6.20).
PROOF OF THE COROLLARY. We shall prove just the case where T =
1/2 since the general case follows from making the change of variables x ~
(2r)-^1 /^2 x. Let¢ be defined by f = lxt -2log¢, so that e-f = e-lxl2
/^2. ¢^2