1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

2. EXISTENCE OF THE HEAT KERNEL ON A CLOSED MANIFOLD 233

as t--+ 0, uniformly for allx, y EM.

EXERCISE 23.18. Show that for k, i 2'.: 0,

(23.65) laf\7~wNI (x, y, t) = o (tN+i-~-k-^2 £)

as t --+ 0, uniformly for all x, y E M.

PROOF. Let

uk (x, y) ~ rJ (d (x, y)) <fak (x, y).

By (23.64) and (23.59),

00

WN = H -PN = PN * 2..::: (D:1PN)*k.

k=l

Moreover, by (23.62), we have

(PN * ~(D,Pw)'•) (x,y,t)

rt CVol(M)(t-s)N+l-~ r

sC Jo (t-s)N-~e N+l-2 JM[PN(x,z,s)[dμ(z)ds

rt CVol(M)(t-s)N+l-~

S canst Jo (t - s)N-% e N+i-~ ds

CVol(M)t N -2 n+l t N+l -2 n

<canst e N-~+i

- N+l-1! 2

for 0 < t S 1 since JM [PN (x, z, s)[ dμ (z) S canst. Hence we obtain the
desired estimate for w N. D

2.4. Convergence of the parametrix convolution series.

Let PN be the parametrix in (23.50). Consider the series l..::~ 1 (DxPN )*k.
From (23.11), (23.38), and the proof of Proposition 23.12, we can show the
following.

LEMMA 23.19. We have

(23.66)

NA

DxPN (x, y, t) = t E (x, y, t) qN (x, y, t),

where

A _:fr ( d

2

E (x, y, t) ~ ( 47rt) 2 exp - (x,y))

5

t

and qN E C^00 (M x M x [O, oo)).