- EXISTENCE OF THE HEAT KERNEL ON A CLOSED MANIFOLD 237
From this we see that for F satisfying the above hypothesis and k 2:: 2,
we haveof 01; ( Fk) (x, y, t) = ( (of o'{; F) F*(k-I)) (x, y, t).
The derivatives of the series (23.55) absolutely converge locally uni-
formly.
LEMMA 23.23. Let K be any compact subset of any local coordinate sys-tem (U, {xi}) and let T < oo. Given£, m E NU{O} and any N > ~+2£+m,
the series 00'Lofor;i ((DxPN)*k)
k=l
converges absolutely and uniformly on JC x K x [O, T]. Hence
00ofo;"GN = 'Lofor;i ((DxPN)*k)
k=l
exists and is continuous, where G N is defined in (23.55).
PROOF. Recall from (23.43) that(23. 76) Ut !1£'7m v x (D x p N ) ( x' y' t) = tN-Il-2£-m 2 e _ d2(x,y) St G m,£ ( x' y' t) '
where Gm , e is a C^00 covariant k-tensor on M x. M x [O, oo. ). In particular,
limofV'~(DxPN) (x,y,t) = 0
t~O ..
provided 2£ + m < N - ~· Thus, by (23.75) we have(23.77) ofo'{; ((DxPN)k) = (ofo1;DxPN) (DxPN)*(k-I).
Now (23.76) implies there exists Cm,£< oo such thatI ( ofo1;DxPN) (x, z, s)I ::::; Cm,£SN-7#;-^2 £-me-d
2
~:·")on JC x K x [O, T]. Thus, from applying (23.69) to the RHS of (23.77), we see
that
lof 01; ( (DxPN )*k) I (x, y, t)
= I lat JM ( ofo1;DxPN) (x, z, s) (DxPN )*(k-l) (z, y, t - s) dμ (z) dsl
ck-I Vol (M)k-2 t(k-l)(N-7#;+1)-l
< ~~~~~~~~~----o--::-~-- (k - 2)! (N - ~ + l)k-^2
X llat JM Cm,£SN-7#;-2£-me_d2~:·") e-~] dμ(z)dsl
C ck-lt(k-l)(N-7#;+1)-ltN-7#;-2£-m+I dz(x,y)
< m,£ Vol (M)k-l e--st-.- (k - 2)! (N - ~ + l)k-^2 (N - ~ - 2£ - m + 1)