312 25. ESTIMATES OF THE HEAT EQUATION FOR EVOLVING METRICS
Summarizing, we have proved for 0 < r' < r :'S 2ro that
(25.28)
where the LP-norms are with respect to the product measure dμ9 dr, i.e.,
for a space-time region P and a function f defined on P,
1
//f//Lq(P) ~ (Jl fqdμ9 dr) <i.
That is, for the subsolution v, we have estimated the higher L n~^22 P-norm
in the smaller parabolic cylinder Pr' in terms of the lower L^2 P-norm in the
larger parabolic cylinder Pr.
STEP 3. Bounding the L^00 -norm by the L^2 -norm. We now iterate this
estimate to obtain an L^00 -norm estimate. In particular, when considering
the L^2 P-norm in (25.28), for some p 2: 1, we shall choose r, r', and 'lj; to
depend on p. For i E N define (in view of r' < r in (25.28))
(25.29) Ti ~ ro ( 1 + 2 i~l)
(so that Ti is decreasing in i, r1 = 2ro, and limi-+oo Ti = ro) and, motivated
by (25.28), define
Pi=;=. (n --+ 2)i-1
n
(so that Pl = 1 and limi-+oo Pi = oo).
Corresponding to ri (and Pi), we shall now construct space-time cutoff
functions
'l/Ji : n x [0, T] --+ [0, 1]
for all i EN. Define a 000 function <Pi: [O, oo)--+ [O, 1] so that
¢i = { 1 on [O,n+il,
0 on h, oo)and
2i+l
---:'S ¢~ :'S 0
ro
(note that Ti - Ti+I = ;n. Define a 000 function 'T/i : [O, T] --+ [O, 1] so that
(25.30)
and
'T/i = { 1 on [ro - rf+u T],
0 on [O, ro - r[J(note that ri^2 - ri+l^2 2: r2
2 i~^1 ). Let
(25.31) 'l/Ji (x, r) ~ ¢i (d9 (x, xo)) · 'f/i (r).