- EXISTENCE OF THE HEAT KERNEL ON A CLOSED MANIFOLD 235
converges for all t E lR+, we have the following.
COROLLARY 23.21 (Convergence of the parametrix convolution series).
The series
00 2
L...J ~ e d cx,y) 5t · (D x p N )*k
k=lconverges absolutely and uniformly on M x M x (0, T] for any T < oo.
Hence the series
00
L (DxPN)*k
k=l
converges absolutely and uniformly on M x M x [O, T] for any T < oo.We consider the convergence of the derivatives in the next subsection.PROOF OF LEMMA 23.20. We shall prove by induction that for any k E
N we have (23.69).
(1) k = 1. By (23.66) we have
N n d2(x,y)
(23.71) IDxPNI (x, y, t) S Ct -2 e--5t-on M x M x [O, T], where C is given by (23.68).
(2) Suppose that (23.69) holds for some k 2 1. We proceed to show this
estimate for k replaced by k + 1. Note that^7
d^2 (x, z) d^2 (z, y) (d (x, z) + d (z, y))^2 d^2 (x, y)
(23.72) s + t - s 2 t 2 t.
Using this, (23.46), ( 23. 71), and the induction hypothesis, we compute
that
I (DxPN )*k+l (x, y, t) I
=\lot JM (DxPN)*k(x,z,s) DzPN(z,y,t-s)dμ(z)ds\
< rt r Ck Vol (M)k-l sk(N-~+1)-le-~ IDzPN (z, y, t - s)I dμ (z) ds
- j o j M ( k - 1)! ( N - ~ + 1) k-l
k ( n ) d^2 (a: z) N n d(^2) (z,y)
1
t 1 ck Vol(M) -lsk N-2+1 -^1 e----y;;--C(t - s) -2 e-5(t-s)
:::; dμ(z)ds
o M (k - 1)! (N - ~ + l)k-l
ck+l Vol (M)kt(k+l)(N-~+1)-le-d\~·Yl
< - kl (N - ~ + 1) k '(^7) For x, y E JR and a, b > 0 we have "': + Yb^2 ;:::: (a:a":t ; indeed, multiplying this by a+ b