1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

5. HEAT KERNEL ON NONOOMPAOT MANIFOLDS 297

and

l

_a -_a I:::; VT-0'.

8vy,T 8vy,CT

REMARK 24.37. Also note that by (24.93c) we have for a E (!, 1) that

r dO' f IM1I (xo, T; z, O') dμg(O') (z) :::; CTl-a

Jo 1aM

is finite. This implies

IA1I (xo, T) :S CT^1 -a.

REMARK 24.38. Note that (24.93a) and (24.93b) follow directly from

Lemma 24.35.

(2) k 2:: 2. Recall that a E (!, 1). We estimate !Mk! by a type of

induction. Suppose we have for some k E N an estimate of the form

(24.94) IMkl (xo, T; z, O') :::; ck (T - O')-f3k d;'Yk (xo, z)

for some Ck < oo, where f3k < 1 and 'Yk :Sn - 2a. Note that by (24.93c),

for k = 1 we have (24.94) with C 1 = C#, /3 1 = a, and 'Yl = n - 2a.
Let Co E [1, oo) be such that
(24.95) C 01 g (T1) :S g (T2):::; Cog (T1)

for all T1, T2 E [O, T]. Recall also that for any a, /3 < 1,

1

T ( T - p )-a ( p-0' )-(3 d p-- ( T-0' )1-a-(311 (1 -p -)-a p --(3d-p

(}' 0

(24.96) = Ca,(3 (T-O')l-a-(3,

where

(24.97)

c --'---r (1 - a) r (1 - /3)

a,(3 .... r (2 - a - /3)

and r is the Gamma function. Thus, by applying (24.93c) and (24.94) to
(24.89), we obtain
IMk+l (xo, T; z, O')I
(24.98a)

(24.98b)

8H

(xo, T; w, p) !Mk! (w, p; z, O') dμ 9 (p) (w) dp

8vw,p

:S 2C#Ck 1

7

{ (T - p)-a d;n+^2 a (xo, w)

(}' JaM

x (p - O')-f3k d-;;'Yk (w, z) dμg(p)(w)dp

< - CC k C a, (3 k Q'Yk/2 O (T - O')l-a-f3k

x f d;n+^2 a (xo, w) d';'Yk (w, z) dμ 9 ( 7 ) (w),

JaM