1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

4. LOGARITHMIC SOBOLEV INEQUALITY 249

PROOF OF THEOREM 6.39 FROM THE PROPOSITION. Given f such that

(6.74) r (27r)-n/^2 e-f dx = 1,

}ffi.n

let 't/J ~ (27r)-n/^4 e-11^2 , so that log'i/J = -f -~log (27r) and fffi.n 1j;^2 dx = 1.

Then (6.73) implies

(6.75) ~log (-

2

- f ~IV' 112 e-t (27r)-nl^2 dx)

4 1fen }ffi.n 4

2 - [n (~+~log (27r)) e-f (27r)-n/^2 dx,

so that

~ f IV' fl2 e-t (27r)-n;2 dx 2 en exp{-~ f fe-t (27r)-n;2 dx}.

2 }"!Rn 2 n }w,_n

We claim

(6.76) en exp{-~ r fe-f (27r)-n/^2 dx} 2 r (n - f) e-f (27r)-n/^2 dx,

2 n }w,_n }w,_n

which implies the T = 1/2 case of (6.69):

(6.77) [n (~IV' fl^2 + f - n) e-f (27r)-n/^2 dx 2 0.

Since fw,_n (27r)-n/^2 e-f dx = 1, inequality (6.76) is equivalent to

~exp{~ r (~ -!) e-f (27r)-n/^2 dx} 2 ~+ r (~ -1) e-f (27r)-n/^2 dx.

2 n }ffi.n 2 2 }w,_n 2

If we let du = e-f (27r)-n/^2 dx and g = ~ - f, then the above inequality
becomes
exp { ~ r gdu} 2 1 + ~ r gdu.
n }w,_n n }ffi.n
This follows from ea 2 1 + a for all a E R D

Now we present the

PROOF OF PROPOSITION 6.43. By Jensen's inequality, if fw,_N !Fir dx =

1, then

(p - r) f (log IFI) !Fir dx = f log (IFlp-r) !Fir dx

}w,_N }w,_N

:S: log ([N IFlp-r JFlr dx) = log ([N IFIP dx)

for p 2 r > 0. The L^2 -Sobolev inequality says

llFlliN* :::; AN r l\7 FJ

2

}w,_N dx,