242 23. HEAT KERNEL FOR STATIC METRICSso that d~ /i = a~i. By the mean value theorem, we have
(23.91)
l; JN (ri (h), y, s, t) ds - l; JN (ri (0), y, s, t) ds it 8JN ( ) d- -a. x,y,s,t s
h 0 ~
rt (8JN 8JN )
=lo axi (ri (h*), y, s, t) - axi (x, y, s, t) ds
for some h* contained in the interval from 0 to h. Note that, in regards tothe integral on the RHS of (23.91), by (23.89) we have that for any c: > 0,
there exists 5 ( c:) > 0 such that
rt I aaJ~ (x', y, s, t) Ids< c:,
lt-o(i;) xindependent of x' E M. On the other hand, given 5 (c:) > 0, there exists
'Tl> 0 such that if lhl <'fl, then
'
aJN - aJN I c:
8. (ri (h*) , y, s, t) - -
8
. (x, y, s, t) < -
~ ~ t
for s E [O, t - 5 (c:)] since we are away from the singularity of PN. We con-
clude from splitting the integral on the RHS of (23.91) as l; = l;-o(i;) + lLo(i;)that if lhl <'fl, then
l; JN (ri (h) 'y, s, t) ds - l; JN (Ii (0) 'y, s, t) ds
hrt aJN I
- lo axi (x, y, s, t) ds < 3c:.
Taking the limit as h--+ O, we obtain (23.90). This completes the proof
of Lemma 23.26. 03.3. Second space derivatives of a convolution with the para-
metrix.
The second partial derivatives of a convolution with the parametrix PNare given by the following (we obtain the same answer as (23.73) with m =
2).
LEMMA 23.27 (Second space derivatives of a convolution with the para-metrix). Under the same hypotheses as Lemma 23.26, for x EU such that
B (x,~inj(g)) cU,
we have (PN * G) (x, y, t) is C^2 with respect to the space variable x and the
second space derivatives of PN * G are given by
(23.92)
a2 1t1 a2pN
a xi ·a xJ · (PN*G)(x,y,t) = o M a xi ·a xJ. (x,z,t-s)G(z,y,s)dμ(z)ds,