238 23. HEAT KERNEL FOR STATIC METRICS
Hence
00
L of a;: ( (DxPN )*k) (x, y, t)
k=l
converges absolutely and uniformly on K x K x [O, T]. D
From the above lemma we also obtainLEMMA 23.24. Given T < oo, f, m EN U {O}, and N > ~ + 2f +m, the
series of covariant m-tensors
00
L af\7~ ( (DxPN )*k)
k=l
converges absolutely and uniformly on M x M x [O, T]. Hence
00
8f\7~GN = L:af\7~ ((DxPN)*k)
k=l
exists and is continuous.3. Differentiating a convolution with the parametrix
In the last section, when applying the convolution to the proof of the
existence of the heat kernel (see the derivation of (23.51)), we used some
basic properties about differentiating a convolution with the parametrix. In
this section we present these properties; an issue in proving them is that the
parametrix in (23.50) is singular when t - u = 0.
3.1. Continuity of integration against the parametrix.
Let (Mn, g) be a closed Riemannian manifold and let PN be the paramet-
rix defined by (23.50). Let IP&f be as in (23.30).
LEMMA 23.25 (Continuity of integration against the parametrix). If f:
M x [O, T] -+IP& is a continuous function, where T E (0, oo), then
IN : M x IP&f -+ IP&,
defined byIN (x, t, u) ~JM PN (x, z, t - u) f (z, u) dμ (z),
is a continuous function on all of its domain M x IP&f and
(23.78) lim IN (x, t, u) = f (x, u)
t"-.,,u
uniformly with respect to x and u. Moreover, IN is C^00 with respect to x
and t; for any f and m,
(23.79) afa;: IN (x, t, u) =JM afa;: PN (x, z, t - u) f (z, u) dμ (z).