316 25. ESTIMATES OF THE HEAT EQUATION FOR EVOLVING METRICS
since llvllL=(PPf) < llvllL=(P2ro) < 00 independent of e. Provided we can
show that
.e-1 A 2-k
(25.45) lim sup II (ck) ::; c
C-too k=O
for some constant C < oo, we obtain from taking the lim sup as f,--+ oo of
the RHS of (25.44) that
(25.46) llvllL=(Pro) ::; C llvl1Ll(P 2 r 0 ) •
Now by (25.42) we have
(
m A 2-k m eCs ( 1+4VKro ) '.;j'c (n+3)n 4 6 n ) 2-k
limsup IT (ck) ::; lim II^0 1 n2(k+^2 )4
m-too k=O m-too k=O ro (Vol_g B_g (xo, ro))^2
(eCs(1+4VKro)'.;j'6~n~3)n Cn) 2 Cn.
(25.47) ::; 2 ) =;= c,
r 0 Vol_g B_g (xo, ro
where Cn ~ 2'.;j' I:k'=o(k+^2 )^2 -k < oo; this yields (25.45).
Finally, since (25.10) says u = eA-r v, by (25.46) we have
(25.48)
sup u::; eATo llvllL=(.P. )
P9 ( xo,To,ro,-r5) ro
::; CeATo llvllv(P 2 r 0 )
::; CeATo llullu(P 2 r 0 )'
where C < oo is given by (25.47). Tracing through the dependence of C
and A completes the proof of Theorem 25.2.
1.3. Summary of the proof of the parabolic mean value inequal-
ity.
Consider the special case of a subsolution v of the heat equation on a
fixed Riemannian manifold with Re?:: 0. Since g~ ::; ~v, for any p E [1, oo)
we have
{)
or ( vP) - ~g(T) ( vP) ::; 0.
Given a cutoff function 'ljJ with support in a compact domain D x [O, T], via
integrating by parts the above inequality against vP, we obtain for 0 ::; r 1 <
T2 ::; T