- MATRIX DIFFERENTIAL HARNACK ESTIMATE
Hence( :t -~) (~Rai3 + RaiJ·y'J~o)
1
= 2 (\7 ;y \7-yRapRpi] + \7-y \7 ;yRpi]Rap) + 2\7-yRap \7 ;yRpiJ
1+ 2Ro;y (\78\7-rRai3 + V-y \78Rai3) - ~ (Ra;yR-yiJ)
{)
+ (V-r \78Rai3 - RapRpiJ-yJ) ~o + Rai3-r8 ot R;yo
+ 2Rai3-r8RopRp;y.Finally we obtain (2.122) from the commutator equations:
and
1 1
2 (V-y\78Rai3 - \7-yVJRaiJ) = -2 (R-yJcxpRpiJ -R-y8pi]Rap)\7 ;y \7-yRap - \7-y\7 ;yRap = - (~-yaifRcrp + ~-ypcrRaa)
= RaaRcrp - RupRaa = O,
which imply
LEMMA 2.95.( :t -~) (\7 -rRai3x-r)
1
= -2 (Rap \7 -rRpiJ + \7 -yRai5RpiJ + \7 pRai3R-rli) x-r- (\7 pRaaR-yiJcriJ - Raa-yp \7 pRui3) x-r + \7-y ( Rai]paRup) X-Y
(2.125) + \7-rRai3 (:t -~) x-r - \78\7-rRaiJ\7oX-Y - \7o\7-rRai3\78X-Y.
PROOF. We compute using (2.105)( :t -~) \7-yRai3 = \7-y ( :t -~) Rai3 + (\7-y~ - ~ V-y) Rai3
- ( :t r~a) RpiJ
= \7-y (Rai]pJRop - RapRpi]) + gPa\7-yRaaRpiJ
1(2.126) + 2 (-RcriJ\7-yRaa +Raif \7-yRui])
1- R-ypaif \7 pRcriJ + R-ypcri] \7 pRaa - 2 R-ya \7 uRai3·
117D