234 23. HEAT KERNEL FOR STATIC METRICS
PROOF. Since qN E c^0 (Nl x M x (O,oo)) and qN(x,y,t) is C^00 in x
and t, the issue is at t = 0. Note that, regarding the first term on the RHS
of (23.38), 'f/OxHN E C^00 (M x M x [O,oo)). From (23.9) and
~ k r 8
'\lxHN = E ~ '\lx<f>kt - 2 t 8 r · HN,
k=Owe have that the last two terms on the RHS of (23.38) may be expressed as
(23.67) (flxTJ) HN + 2 ('! x'f/, '! xHN)
A ( d2
= E ·exp (x,y))~( / (8</>k r )) k
20 t f;:a (flxTJ) </>k + 2'f/ · 8 r -^2 t</>k t ,which has support in (MiniJg) - Minilg)) x (0, oo). Now the fact that
qN (x, y, t) in (23.66) is c^0 at t = 0 follows from the easy fact that for
any M E N there exists C < oo independent of t such that
(
d2
exp (x,y))~(^1 (8</>k r )) k
20 t ~ k=O (flxTJ) </>k + 2'f/ · 8 r - 2 t </>k t< C exp (-inj2 (g)) (cl+ tN)
- 320t
::::; CtM
on ( MiniJg) - Minjlg)) x (0, 1), where in the last line the dependence of
C < oo includes M. Finally, that qN (x, y, t) in (23.66) is C^00 at t = 0
follows from similar estimates for the derivatives of equation (23.67). D
We now proceed to show that (23.66) is sufficient to imply that the series
(23.55) converges.
LEMMA 23.20 (Estimates for the convolutions of DxPN).andT E (O,oo), let
(23.68) C ~ sup JqNJ < oo,
MxMx[O,T]GivenN > n/2
where qN E C^00 (M x M x [O, oo)) is as in (23.66).
have
Then for any k E N we(23.69)
As a consequence of the lemma, since the series
(23.70)
(^00) Ck Vol (M)k-l tk(N-~+l)- (^1) N-T! (CVol (M) tN-~+l)
L n k-1 = Ct 2 exp n
k=l (k-l)!(N- 2 +1) N-2+1