3. DIFFERENTIATING A CONVOLUTION WITH THE PARAMETRIX 243
where we assumed that {xi} are geodesic coordinates centered at x.
Since b. = gij Bx~;xj at the center of geodesic coordinates {xi}, an im-
mediate consequence of Lemma 23.27 is the following.
COROLLARY 23.28 (Laplacian of a convolution with the parametrix). If
GE c^0 (M x M x [O, oo)), then
(23.93)b.x (PN * G) (x, y, t) =lot JM b.xPN (x, z, t - s) G (z, y, s) dμ (z) ds.
PROOF OF LEMMA 23.27. By (23.79) we have(23.94) 82JN^1 32pN
8.
8
. (x,y,s,t) =
8
.
8
. (x,z,t-s)G(z,y,s)dμ(z).
xi xJ M xi xJ
Now differentiating (23.87), we have
82 PN
(^8) xJ '8 xi. (x,z,t-s)
(
1 8 (r;)) ( 8ry T/ 8 (r;)) ~ k
=E 4(t-s) 8xj 8xi - 4(t-s) 8xi f::o¢k(x,z)(t-s)
(
+E - (^1 )-8(r;)) LN 8¢k k
8-. T/ -
8
4 t - s xJ xi. (x,z)(t-s)
k=O(327/ 1 8ry 8 (r;)) LN k
+E (^8) xJ ·3. xi - 4( t - s )-8 xJ .-8-. xi
k=O
(
l 82 (r;)) N k
+E ( )T/ 8.
84 t - s xJ xi. ""'<Pk(x,z)(t-s) L..t
k=O(
8ry 'f/ 8 (r;)) N 8¢k. k
+E 8xi -. - 4 ( t - s )- 8 -. xi ""'-L..t 8 xJ. (x,z)(t-s)
k=O- E ( :::; t, ~!~ (x, z) (t - s)' + ~ t, /)~!:, (x, z) (t - s)').
From this we may estimate
\ ::j~:i (x, z, t - s)\< C (i + 1 + r; (x) ) (t - s)-n/2 exp (- r; (x) )
- t-s (t-s)^2 4(t-s)
< C (t _ s)-1 r-n (x) (i + r; (x)) (r; (x)) ~exp ( r; (x) )
- z t-s t-s 4(t-s)