1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

3. DIFFERENTIATING A CONVOLUTION WITH THE PARAMETRIX 249

for s E [O, t - 8 (c)]. We conclude from splitting the RHS of (23.108) as

Jo rt = Jo rt-o(c:) + Jt-o(c:) rt t h at i "f I I h < 'fl, t h en

J~ ~ ( "'/i ( h) , y, s, t) ds - J~ ~ (Ii ( 0) , y, s, t) ds

h

1

• t EPJN I
  8

.

8

. (x,y,s,t)ds < 3E.

o xi xJ

Taking the limit as h-+ O, we obtain (23.107). This completes the proof of

Lemma 23.27. D

3.4. Time derivative of a convolution with the pararnetrix.

Now that we have formulas for the first two spatial derivatives of a
convolution with a parametrix, we consider the time derivative (compare
with (23.74)).

LEMMA 23.29 (Time derivative of a convolution with the parametrix).

If G E C^0 (M x M x [O, oo)), then PN * G is C^1 with respect to the time

variable and

(23.109)

{)

ot (PN * G) (x, y, t)

= G (x, y, t) +lat JM it PN (x, z, t - s) G (z, y, s) dμ (z) ds.

PROOF. Recall from (23.85) that

(PN * G) (x, y, t) =lat JN (x, y, s, t) ds,

where

JN (x, y, s, t) =JM PN (x, z, t - s) G (z, y, s) dμ (z).

The time-difference quotient associated to PN * G is

(PN * G) (x, y, t + h) - (PN * G) (x, y, t)

h

(^1) ( rt+h rt )
= h J
0
JN ( x, y, s, t + h) ds - J
0
JN ( x, y, s, t) ds
- 11t+h JN ( x,y,s,t+ h)d s+ 1tJN(x,y,s,t+h)-JN(x,y,s,t)d h s
h t 0
= -11t+h JN ( x, y, s, t + h) ds + 1t ~JN {) I ( x, y, s, t) ds,
h t 0 ut t=t*