- LOG SOBOLEV INEQUALITY VIA ISOPERIMETRIC INEQUALITY 209
to Mt with H = 1~~1-l and f =~'we have
F (t) = Vol 0 (Mt) = f
00
( { 1~~1-l do-) ds,lt J{~=s}
where dO-is the volume (n - 1)-form of {x EM:~ (x) = s}, and- ddF (s) = { 1~~1-l dO- for a.e. s > 0.
S }{~=s}
Hence we have(22.96)
1-^00 A. (s) -d dF (s) ds =^100 A. (s) J 1\1~1 _ -1 dO-ds.
o^8 o N=s}
Again by the co-area formula
J 1
A. ( ~) dμ =^00 A. ( s) J 1-\l ~ 1-1 dO-ds.
{xEM:1fi(x)>O} 0 {~=s}(22.97)
Combining (22.95), (22.96), and (22.97), we have1
00
dA. d ( s) F ( s) ds = J A. ( ~) dP,.
O S {xEM: 1fi(x)>O}For the same reason, we have (see Exercise 22.18 below)
(22.98)
100
-d dA. (s) F (s) ds = J A. (h) dμJE.n,
0 S · {yEJE.n: h(y)>O}and hence
(22.99) f A. ( ~) dP, = f A. ( h) dμJE.n.
l{xEM: ~(x)>O} }{yEJE.n: h(y)>O}Choosing A. ( s) = (log s^2 ) s^2 + sns^2 , where Sn = ~ log(27r) + n, the above
inequality implies
JM ((1og~^2 ) ~^2 + Sn~^2 ) dP,
= r ((1og~^2 ) ~^2 + Sn~^2 ) df;,
j{xEM :~(x)>O}= r. ((log h^2 ) h^2 + Snh^2 ) dμJE.n
}{yEJE.n: h(y)>O}= Ln ((log h^2 ) h^2 + Snh^2 ) dμJE.n.
Note that we have JM ~^2 dP, = fJE.n h^2 dμJE.n by choosing A. (s) = s^2 in (22.99).
Claim.