- DIFFERENTIATING A CONVOLUTION WITH THE PARAMETRIX 239
We leave the proof as an exercise or see Theorem 1 on pp. 4-6 of [61]
where the analogous result is proved for bounded domains in Euclidean
space.
3.2. First space derivatives of a convolution with the para-
metrix.
In the next two lemmas we shall show that we can differentiate in space
(at least up to order two) a space-time convolution with a parametrix under
the integral sign. The first partial derivatives of a convolution with the
parametrix PN are given by the following (we obtain the same answer as
(23.73) with m = 1).
LEMMA 23.26 (First space derivatives of a convolution with the para-metrix). Let (U, {xi} ~= 1 ) be a local coordinate system on M. If
GE c^0 (M x M x [O, oo)),
then PN * G is C^1 with respect to the space variables and for x E U, the first
space derivatives of PN * G are given by
(23.80)~^8 (PN * G) (x, y, t) = 1t1 EJPN ~ (x, z, t - s) G (z, y, s) dμ (z) ds.
uxi o M uxi
PROOF. Recall from (23.35) and (23.9) that
N
(23.81) PN (x, y, t) ~ rJ (d (x, y)) E (x, y, t) L <fak (x, y) tk,
k=Owhere E (x, y, t) = (47rt)-n/^2 exp (-r; 4 ~l) and ry (x) ~ d (x, y). In particu-
lar,
(23.82) (PN * G) (x, y, t) =lat JM PN (x, z, s) G (z, y, t - s) dμ (z) ds
is an improper integral, with the integrand having a singularity at (z, s) =
(x, O). We address this issue below while showing that the first space deriva-tives of PN * G exist.
For any n 2:: 2, r, s > 0, and a E (0, 1) we have
(
s-n/2 exp -4s r2) = s-ar2a-n (r2)~-a --; exp ( - r2) 4s< _ C s r -a 2a-n ,
so that
(23.83) PN (x, z, s):::; CE (x, z, s) :::; Cs-arx (z)^2 a-n,
where C < oo is independent of x, z E M and s > 0. Note also that
IGI is bounded on compact sets and PN (x, ·, s) has compact support in