- EXISTENCE OF THE HEAT KERNEL FOR A TIME-DEPENDENT METRIC 273
2.2. The parametrix convolution series.
Similarly to the previous chapter, we shall construct the heat kernel bya convolution series. Recall that IRS, = { ( T, v) E IR^2 : T > v}.
DEFINITION 24.13 (Space-time convolution). Given two functions
H, J E c^0 (M x M x IRS,),their space-time convolution is defined by
(24.28) (H * J) (x, T; y, v) ~ 1
7
JM H (x, T; z, (}) J (z, (}; y, v) dμg(CT) (z) d(}as long as the integral is well defined.
REMARK 24.14. Note that when g ((}) = g, by letting u = 0 and by tak-
ing H (x, t; z, s) = F (x, z, t - s) and J (z, s; y, u) = G (z, y, s - u) in (24.28),
we have
(H * J) (x, t; y, u) ~lat JM F (x, z, t - s) G (z, y, s) dμ 9 (z) ds,
which is the same as (23.46).
LEMMA 24.15. The space-time convolution in (24.28) is associative.
PROOF. We compute
(24.29)((H J) K) (x, T; y, v)
= 1
7
JM (H * J) (x, T; z, (}) K (z, (}; y, v) dμg(CT) (z) d(}=1T1T r H(x, T; w, p) J(w, p; z, (}) K(z, (}; y, v)dμg(p)(w)dμg(CT)(z)dpd(}
v CTJMxM ·and
( H (J K)) ( x' T; y' v)
= 1
7
JM H (x, T; z, (}) (J * K) (z, (}; y, v) dμg(CT) (z) d(}=1T1CT r H(x, T; z, (}) J(z, (}; w, p) K(w, p; y, v)dμg(p)(w)dμg(CT)(z)dpd(}.
v vJMxM
Exchanging the order of integration with respect to p and (} (and switching
the labels w and z), we obtain
(H (JK)) (x,T;y,v)
=1
7
r r H(x, T; w, (}) J(w, (}; z, p) K(z, p; y, v)dμg(p)(z)dμg(CT)(w)d(}dp
v }p JMxM=1T1T r H(x, T; w, p) J(w, p; z, (}) K(z, (}; y, v)dμg(CT)(z)dμg(p)(w)dpd(},
v CT JMxM
where in the last line we relabelled p and(}"; this is the same as (24.29). D