- EXISTENCE OF THE HEAT KERNEL ON A CLOSED MANIFOLD 229
their (space-time) convolution is given by(23.46) (F * G) (x, y, t) ~lat JM F (x, z, s) G (z, y, t - s) dμ (z) ds
as long as the integral is well defined. Clearly ( F, G) f---t F * G is bilinear.
Since we are assuming· M is closed, all of the integrals regarding con-
volutions of c^0 functions on M x M x [O, oo) are finite. When one of the
functions in the convolution is a parametrix, the finiteness of the resulting
integrals and their first and second derivatives is an issue which we shall
address.
The convolution operation is associative; that is, if we are given three
functions F, G, HE c^0 (M x M x [O, oo)), then(23.47) (F * G) * H = F * (G * H).
Indeed, we compute
(23.48)( ( F G) H) ( x, y, t)
= it JM ( F * G) ( x, z, s) H ( z, y, t - s) dμ ( z) ds
=it ls JMJMF(x,w,r)G(w,z,s-r)H(z,y,t-s)dμ(w)dμ(z)drds,
whereas
(23.49)( F ( G H)) ( x, y, t)
=it JMF(x,w,v)(G*H)(w,y,t-v)dμ(w)dv
=lat lt-v JM JM F(x, w, v) G(w, z, u) H(z, y, t - v - u) dμ(z) dμ(w) dudv
= ft f8 f f F(x, w, s - u) G(w, z, u) H(z, y, t - s) dμ(z) dμ(w) duds
Jo Jo JMJM ·
=lat ls JM JM F(x, w, r) G(w, z, s - r) H(z, y, t-s) dμ(z) dμ(w) drds.
The associativity formula (23.47) now follows from comparing (23.48) and
(23.49) while using Fubini's theorem to change the order of integration with
respect to the variables w and z.
Since convolution is associative, given F E c^0 ( M x M x [O, oo)) and
k EN, we may uniquely define
p*k ~ F * .. · * F (k many F's),