222 23. HEAT KERNEL FOR STATIC METRICS
By induction on k ::::;: 0, we see that the functions k defined recursively
by (23.26) are C^00 on Minj(g) for all 1 :::; k :::; N. This completes the proof
of part (2) of Lemma 23.7.
From (23.26) we see that along the diagonal of Minj(g)(23.27)
1
for 1:::; k:::; N. Note that formula (23.27) also follows directly from (23.23b)
when assuming the finiteness of </>k and l'V</>kl·^5
Now that we have constructed a good approximation, in the next sub-
section we establish some of its properties.
1.3. Properties of the good approximation to the heat kernel
- bounds on its heat operator and derivatives.
Let
8Dx ~ b.x - at·
Since E restricted to Minj(g) x IRS, is C^00 and since </>k : Minj(g) -+JR is C^00
for 0 :::; k:::; N, we conclude from definition (23.9) that
N
HN =EL </>k · (t - u)k E C^00 (Minj(g) X JR;).
k=OFurthermore, since </>N E C^00 (Minj(g)), by (23.11) and N > n/2, we may
extend the function DxHN, which is defined on Minj(g) x lRS,., continuously
to a function DxHN defined on Minj(g) x JR$ and taking the value 0 on
Minj(g) x 8 (IRS,). In fact, by (23.11), since lb.x
(23.28)
where N - ~ > 0.
Next we consider bounds for the space-time derivatives of DxHN. Let
(U, {xi} ~= 1 ) be any local coordinate system in M. Let 8~ = Ox o · · · o Oxdenote some k-th partial derivative in these coordinates and let of ~ gt~
denote the £-th time derivative fork,.€ EN U {O}. Differentiating (23.11),
we have in ((U x M) n Minj(g)) x JR$ (not explicitly writing the coefficients
(^5) Note that a model case for the ODE (23.23b) is r ~~ + k¢ = 0, whose solutions are of
the form¢ (r) = Cr-k, where CE R In particular, the only finite solution is¢ (r) =. 0.