- HEAT KERNEL ON NONCOMPACT MANIFOLDS 299
Now suppose(24.102) 2a -1 < /k Sn - 2a.
Regarding the integral in (24.98b), since 8M = R1 U R2 U R3 and by
summing the estimates (24.99), (24.100), and (24.101), we have
(24.103) r d:;n+^2 a (xo, W) d:;"fk (w, z) dμg(T) (w) S cd;CX-"(k-l (xo, z)'
JaMwhere C < oo is independent of k. Thus, assuming (24.94) and (24.102), we
have that (24.98b) implies
(24.104)
where
ck+i ~ cckca,f3k CJ'k^12 ,
-f3k+l ~ 1 - a - f3k,
-/k+l ~ 2a - rk -1.Therefore there exists k EN such that for any 1 S k S k, we have
(24.105)
where either -f3rc ;:::: 0 or -1rc 2:: 1 - 2a, i.e., -/rc+i 2:: 0.
Case 1. There exists ko = k + 1 E N such that -/ko 2:: 0. Then by
(24.104) we have
IMko I (xo, T; z, O') SC (T - 0')-/3,
where f3 = f3ko+l and C = Cka+ 1 diam (g ( T) )-'Yko+^1. Substituting this in
(24.98b), we have
JMko+l (xo, T; z, O')Jwhere
SC 1T (T - p)-a (p - 0')-(3 dp r d:;n+^2 a (xo, w) dμg(T) (w)
~ JaM
< CC'r (1 - a) r (1 - (3) ( _ )l-a-/3- r (2 - a - (3) T O' '
C' ~ sup r d;n+^2 a (xo, w) dμg(T) ( w) < oo.
xoEaM laM
In general, we have for f E N U { 0}
(')£ re (1 - a) r (1 - (3) £(1-a)-/3
IMko+e(xo,T;z,O')lsC c r(1+£(1-a)-f3)(T-O').