236 23. HEAT KERNEL FOR STATIC METRICS
where we also used (t - s)N-~ ~ tN-~ and^8
i
t k(N-1'+1)-l tk(N-~+l)
s 2 ds =.
o k (N - ~ + 1)Estimate (23.69) now follows from induction. D
We leave the reader with the following elementary exercise used in the
proof of Theorem 23.16.
EXERCISE 23.22 (Convolution and sum for DxPN commute). Show that
00 00
(DxPN) * L (DxPN )*k = L (DxPN )*k.
k=l k=2· 2.5. Convergence of the derivatives of the heat kernel series.
We shall show that the sum of the derivatives of the terms of the series
(23.55) converges.
First recall some elementary facts about differentiating convolutions.
Assuming that Fis cm in the first space variable on M x M x [O, oo), by
taking space derivatives of (23.46), we have
8"; (F * G) (x, y, t) =it JM 8"; F (x, z, t - s) G (z, y, s) dμ (z) ds
(23.73) = ((8"; F) * G) (x, y, t).
On the other hand, assuming that F is C^1 in the time variable on M x M x
[O, oo ), by taking a time derivative we have8t (F G) (x, y, t) =JM F (x, z, 0) G (z, y, t) dμ (z) + (8tF G) (x, y, t).
If F (x, z, 0) = 0, then we obtain
(23.74) ot(F * G) (x, y, t) = ((8tF) * G) (x, y, t).
Therefore, given m and £, we have for F differentiable to sufficiently high
order,(23. 75) af a;: (F G) (x, y, t) = ( ( af a;: F) G) (x, y, t)
provided ({)f o;i F) (x, z, 0) = 0 for 0 ~ p ~ £ - 1.
(^8) Another way to estimate (actually, evaluate) the time integrals of the powers of s
and t - s, here and below, is to use the following formula (seep. 15 of [61]). For a, b > 0
we have
1
(^1) (1- )a-1 b-ld = r(a)r(b)
0 a a a r (a+ b) '
where r is the Gamma function. Making the change of variables= ta, this implies
1
t(t- )a-1 b-ld =r(a)r(b)ta+b-1
o s s s r (a+ b).