240 23. HEAT KERNEL FOR STATIC METRICS
B (x, inj (g) /2). Hence, regarding the space integral in (23.82), by (23.83)
and a > 0, we have that
r IPN (x, z, s) G (z, y, t - s)I dμ (z) :::::: ( cs-°'rx (z)^2 °'-n dμ (z)
JM } B(x,inj(g)/2)
tnj(g)/2
:::::: cs-a lo r2a-ldr:::::: cs-°',where we also used Vol(8B(x,r))::::; Crn-l for r E (O,inj(g)/2]. This
implies that the improper integral in (23.82) converges absolutely since
t.
J 0 s-°'ds < oo by a < 1.
Now let(23.84) JN(x,y,s,t) ~JM PN(x,z,t-s)G(z,y,s)dμ(z),
so that
(23.85) (PN * G) (x, y, t) =lot JN (x, y, s, t) ds.
By Lemma 23.25, JN (x, y, s, t) is C^00 in the variable x and continuous in
the variable s.
Let (U, {xi} ~= 1 ) be a local coordinate system and let x E U be as in the
hypothesis of the lemma. By (23.79) we have(23.86) ~ 8JN (x, y, s, t) =^1 - OPN
8. (x, z, t - s) G (z, y, s) dμ (z)
uxi M xi
for x, y EM ands> 0. We shall estimate this integral. Observe that^9
(23.87)
8PN- 8
xi. (x,z,t-s)
(817 rt a (r;)) ;..._ k
= E 8xi - 4 (t - s) 8xi 6 ¢k(x, z) (t - s)
k=O
N
~8¢k k
+ 17E 6 oxi (x, z) (t - s).
k=O(^9) In our estimates, essentially we can choose to ignore the term with :;:; since its
support is away from the singularity.