- MEAN VALUE INEQUALITY FOR SOLUTIONS OF HEAT EQUATIONS 313
Note that
87/Ji 2i 2i+l
(25.32) 0 < - < - and l\77/Jilg-S -.- ar - T5 To
Here \77/Ji is a C^00 vector field on the complement of the cut locus of xo (the
cut locus is a closed set with measure zero).
We then havesupp('t/Ji) C Pri' 7/Ji = 1 on Pri+i"
Moreover, by (25.32) and 7/Ji :::; 1, we have(25.33) 7/Ji ~~i + l\77/Jil~ S 4i+^2 T 02 ~Li.
Now we may take 'ljJ = 7/Ji in Step 2.
Corresponding to (25.27), define
(25.34)M· ='= ecs(i+v.Kri) ((1+1)2 cn+14i+2 + 1) (2cn4i+2T-2) ~
i • Vol2/n ( ) 2i-1 0 0 0 ·
9 B_g xo, Ti
By (25.28), with T =Ti, T^1 = Ti+l, p =Pi, and M =Mi, we have (note that
Pi+l = n~^2 Pi)
n 1
llvllL^2 Pi+l(Pri+i) S Mt+
2 2
Pi llvllL2Pi(Pri)'where we used the fact that Ti :::; 2To and (25.8) imply B (xo, 4To) c D.
Hence, for any integer j 2 2llvllL"'; ( P.,) :S ()j M,.:;,,;,) llv llv (P.>c,) ·
Taking the limit as j-+ oo, we have
llvllL-(P.,) :S (Il M;~' ,;, ) llvllL'(P,.,)
(25.35)
where