1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

3. DIFFERENTIATING A CONVOLUTION WITH THE PARAMETRIX 241

In particular,

(23.88)

I ~~ ( x, z, t _ s) I

"5. C (i + rz (x)) (t-s)-n/2exp ( r; (x) )

t-s 4(t-s)

• _ Crx (z)^2 a-n ((r; --(z)) "¥-a + --^1 (r; --(z)) "¥+1-a) exp ( ----r; (z) )
  ( t - st t - s r x (z) t - s 4 ( t - s)

"5_ C (t - S )-a rx (z)2a-n-I'

where C < oo is independent of x, z EM ands E (0, t] and where we used
the fact that r x ( z) is bounded.
Hence, by (23.88) we have, provided a E (~, 1),

I~ (x,y,s,t)l "5. JM' (x,z,t-s)G(z,y,s),dμ(z)

"5.C {.. (t-s)-arx(z)^2 a-n-^1 IG(z,y,s)ldμ(z)

JB(x,mJ~g))

tnj(g)/2 ·

"5. C (t - s)-a Jo r2a-2dr

(23.89) "5. C (t - s)-a

since supp ( °tx1t (x, ·, t - s)) c B ( x, injJg)), where C < oo is independent

of x, y E M and s E ( O, t]. In particular, the improper integral

i

t1 8PN - it 8JN ·

8

. (x,z,t-s)G(z,y,s)dμ(z)ds= -
8

. (x,y,s,t)ds

o M xi o xi

on the RHS of (23.80) converges absolutely.

By (23.85) and (23.86), proving (23.80) is equivalent to showing that the

partial derivative

(23.90)

• a. a it it aJN
  8
  . (PN * G) (x, y, t) =;= -
  8
  xi xi. o JN (x, y, s, t) ds = o - 8 xi. (x, y, s, t) ds

exists; we do this now. Let 'Yi : (-s, s)---+ M be a smooth path with

'Yi (0) = x,. xj ('Yi (u)) =canst for j i= i,