- EXISTENCE OF THE HEAT KERNEL FOR A TIME-DEPENDENT METRIC 277
The second formula is obvious. The first formula is true since by (24.27) we
have
I
OJN OT (x,T;y,v;O") I I =JM r oPN OT (x,T;z,O")G(z,O";y,v)dμ 9 (<7)(z) I
::::; IJM (Lix,-r - Q) PN (x, T; z, O") G (z, O"; y, v) dμ 9 (<7) (z)I. { N n _d~(x,z)
+Co j M ( T - O") -2 e^5 Cr-a) IG (z, O"i y, v) I dμg(<7) (z)
::::; c ( t - s )-°' + c
for a E ( ~, 1). D
Continuing with our verification of (24.31), we now estimate (Lx,-rPN )*k.
Recall that by (24.27) we have
N n _d~(x,y)
ILx,-rPNI (x, y, T, v) ::::; Co (T - v) -2 e 5(r-v).We shall prove by induction that there exist constants Ck < oo (defined in
(24.39) below) such that on M x M x ~~
(24.37)
for all k EN, where c E (0, 1) is defined in (24.38) below.
Assuming (24.37) holds for a given k, we have
I (Lx,-rPN )*k+l I (x, Ti y, v)
= l(Lx,-rPN) (Lx,-rPN)kl (x,Tiy,v)
c2(k-l)di-(x,y)
5(r-v)= 11
7
JM Lx,-rPN (x, Ti z, O") (Lx,-rPN )*k (z, O"i y, v) dμg(<7) (z) d0"1 ·1
T r n _ di-(x,z)
::::; v }MCo(T-O")N-2e 5(r-a)since v ::::; O" ::::; T,(24.38)
andd(]" (z, y) ?:: c · d 7 (x, z),