196 22. TOOLS USED IN PROOF OF PSEUDOLOCALITY
by ffr H](, R' ::::; 0, by the mean curvature of distance spheres comparison
theorem, ~hich says H (y) ::::; HK (r) (see Lemma 1.126 in [45] for example),
and by ff 7 dg(T) (y, xo) 2: (n - 1) K dg(T) (y, xo). Hence, choosing K' suffi-
ciently large and R' > 0 sufficiently small, both depending only on K, we
obtain
(22.39)and
(22.40) Bg(T) (xo, R') c Bg(O) (xo, R)
for all TE [0,w]. Indeed, without loss of generality we may assume K < 0.
Then we have
HK (r) = (n -1) v1Kf coth ( vlKfr)
= (n-1) (~+ l~lr+O(r^3 )),
so it suffices to take K' = 5K to obtain (22.39) from (22.38) for R' sufficiently
small.
We remark that in the calculation of (22.38) we have used the fact that
for any function f, in spherical coordinates centered at xo, we have
a^21 af
f:::.g(T)f (y) = 8r2 + H (y) ar + f:::.s(p,r)f.We also have
lim H](, R' ( dg(T) (-, xo), T) = Ox 0
T\,0 '
and by (22.40),eL^7 HRl"'B u g(r) ( XQ, R') 2: 0 for TE (O,w].
One can now apply the same argument (a form of the maximum prin-
ciple) as in the proof of Lemma 16.49 in Part II to (22.37) and (22.39) to
obtain
eLT HR (y, T) 2: HK',R' ( d 9 ( 7 ) (y, xo), T)for y E Bg(T) (xo,R') and TE (0,w]. Therefore the adjoint heat kernel Hof
(Mn, g ( T)) centered at ( x 0 , 0) satisfies
(22.41) H (xo, T) 2: HR (xo, T) 2: e-LT H](, R' (0, T).
'
Applying this generally formulated estimate to the adjoint heat kernels
Hi defined by (22.29), we obtain (22.34). This completes the proof of (2).
( 3) Since by ( 1), Hi converges to H 00 > 0 in C^00 , we have h converges
to f 00 in C^00 and the equality H 00 = (47rT)-n/^2 e-f=. Clearly D~H 00 = 0
follows from Di Hi = 0. This also implies f\ converges to v 00 in C^00 and
we have the equality in (22.31). Finally, v 00 ::::; 0 follows from Vi ::::; 0 and
Vi -+ v 00 ; equation (22.32) for D~v 00 follows from taking the limit of the
analogous equations for Divi. This finishes the proof of Lemma 22.9.