- CONSTRUCTION OF THE PARAMETRIX FOR THE HEAT KERNEL 223
in the second line)(23.29)ofo: (DxHN) (x, t; y, u)
e k
=LL coeff ·o{o~E (x, t; y, u) a:-p (b.x</>N (x, y)) a:-q ( (t - u)N)
q=O p=O
= E (x, t; y, u) (t - u)N-k-^2 £ Fk,£ (x, y, t - u)= ( 4 )-n/2 (t _ )N-(n/2)-k-2£ ( d
2
1f u exp (x, Y)) D ( t _ )
4 (t-u) rk,£ x,y, ufor N > ~ + k + 2£, where
Fk,£ E C^00 ( ( (U x M) n Minj(g)) x [O, oo)).
The power N - (n/2) - k-2£ oft - u in (23.29) is due to the fact that each
space derivative of E introduces a (t - u)-^1 factor and each time derivative
of E introduces a (t - u)-^2 or (t - u)-^1 factor. (Recall that d^2 and </>N are
both C^00 functions on Minj(g).)
Given TE (0, oo), let(23.30) lR.f = { (t, u) E JR.^2 : 0 < t - u ST} C JR;.
In particular, formula (23.29) implies that for any k, .e E NU{O}, any compact
subset K c U, and any T E (0, oo ), we have
(23.31)Jaf a: (DxHN) I (x, t; y, u) s c (t - u)N-(n/^2 )-k-^2 £ exp (
S C (t _ u)N-(n/2)-k-2£d^2 (x,y))
4 (t - u)for all (x, t; y, u) E ( (K x M) n Minj(g)) x lR.f. Note that as a special case
(when k = .e = 0), we have (23.28).
The same types of formulas as (23.29) and (23.31) are true on all of
Minj(g) x lR.f, for any T E (0, oo ), with the partial derivatives 8~ replaced
by covariant derivatives 'V~.
LEMMA 23.10 (Covariant derivatives of heat operator of HN)· For any
k,£ENU{O} and TE (O,oo),
(23.32) Jof'V~ (DxHN) J S C (t - u)N-(n/^2 )-k-^2 £ exp (-:~t(~ ~)