- DISTANCE-LIKE FUNCTIONS ON NONCOMPACT MANIFOLDS 385
In particular, we have llf (·, 1)11 0 2,a S C13 in fJ (0, po/2) with respect to
harmonic coordinates. Since\7 \7 8
2rk 8 d rk 1 A kc ( 8 A 8 A 8 A )
i j = 8yi8yj - ij 8yk an ij = 2,g 8yi9jR + 8yj9iR - 8yg9ij '
by (26.149) and (26.150) we have (corresponding to fJ = 0)
l'V'Vfl (x, 1) s C14
for all x E M, where C14 < oo depends only on n and K. This completes
the proof of the proposition. D
Now we give a proof of Remark 26.50. Suppose that (Mn,g) satisfiesl\7k Rml SK for 0 S k Sm,
where m EN. Let p be as in (26.146). By Theorem 6 in Hebey and Herzlich
[97],^14 since l'Vk Rml S K for 0 s k s m, there exists a constant C < oo
depending only on n, K, m, and Po E (0, pj.J]{] such that for any x EM
(with respect to harmonic coordinates (fJi) centered at x)(26.152) 18yi1~~8yir>gpql'18yi1 ~.~;~irn+1gpqlo,a SC
in fJ (0, po/2) for all 0 S /3 S m + 1.
Using this estimate, we can apply the high-order interior estimate of
parabolic Schauder theory to (26.151) and we have
(26.153)in fJ (0, po/4) x rn, l]. However, this estimate is only a local estimate with
respect to partial derivatives in harmonic coordinates. To get an estimate
for all higher covariant derivatives l'V,B JI off, we use induction. By the
.C 1ormu (^1) a rk ij = 29 1 AkC ( ayi (^8) 9j£ A + Bf;j (^8) 9i£ A - aye (^8) 9ij A ) an d b Y
\7· ... \7· f-. 3,B f. = r\7 k-1 f+ (~ ur ) \7 k-21 +·. ·+ (~k-2 u r * ) '7j v
ii if' uy ~ Ai1 ···uy,, ~Ai(./ '
we obtain
I \7,8 f I ( x, 1) S C for 1 S /3 S m + 2
using estimates (26.152) and (26.153). Since x is arbitrary and C is inde-
pendent of x, estimate (26.130) is proved.
REMARK 26.53. By Corollary 4.12 of Hamilton [93] (or Proposition 4.32
in Part I), we have bounds on the partial derivatives, with respect to geodesic
coordinates, of gpq up tom-th order. On the other hand, by DeTurck and
Kazdan [53], harmonic coordinates have maximal regularity.
(^14) In [97] only bounds on the Ricci tensor and its covariant derivatives are assumed.
See also [95].