- A PRIORI ESTIMATES FOR THE PRESSURE FUNCTION 77
We start with a second-order quantity in v. Let <p : 52 ---+JR be any C^3 function.
Recall the ubiquitous Bochner formula(29.26) 6. 9 IV cpl~= 2g (V(6. 9 cp), Vcp) +RIV cpl~+ 21v^2 cpl~,
where we used Re = ~ Rg for dimension 2. From t his we obtain
(29.27) 6.g(R + IV<pl~) =A+ 2g (V(6.g<p - R), V<p) + (6.g<p )^2 - R^2 )
where the first term on the RHS, defined by. 2 IV RI~ IV R + RV<pl~ I 2 1 1
2
A::::;= 6. 9 R + R - ~ + R + 2 V <p - 26.g<pg
9,is nonnegative by the trace Harnack estimate (28.46). Note that (29.27) generalizes
(28.45) in the previous chapter.
The a priori estimates we prove below are uniform on 52 x ( -oo, -1].LEMMA 29.6. For any p E [l , oo) there exists C(p) < oo such that for each
tE(-oo,-1],
(29.28) llv (t)llw2.v :S C(p).For any a E (0, 1) there exists C(a) < oo such that fort E (-oo, -1],
(29.29) llv(t)llc1,a :S C(a).
PROOF. Taking <p = ln v in (29.27) yields6. 9 (R +IV ln vi~) 2: 2g (V(6. 9 ln v - R), V ln v) + (6. 9 ln v)
2- R^2.
Note that IV?/'I~ = v IV?/'l~2 for any function¢. SinceR +IV ln vi~ = 6.52V + 2v
by (29.7), we obtain thatv6.s2(6.s2v+2v) 2:-41Vvl~2 -4vR+4v^2 = -4v6. 5 2v-4v^2.
Define the quantity
(29.30) F ~ 6.52v + 6v,which is constant on the King- Rosenau solution. On 52 x ( -oo, -1] we have
(29.31) 6.s2F 2: -4v 2: -C.
By (29.7), there exists a co nstant C < oo such that 6.52V > -2v 2: -C on
52 x (-oo, -1]. In particular, F > 0. Since IJ 52 Fdμs2 I :::; C, by applying the
mean value inequality to (29.31) we conclude that 0 < F:::; C (see Theorem 4.1 in
[144]). Hence the Laplacian of v is bounded:
(29.32) l6.s2vl :S C.
By applying standard elliptic estimates to (29.25) and (29.32), we see that(29.28) holds for any p E [l ,oo) and t E (-oo,-1]. Finally, the estimate (29.29)
follows from the Sobolev inequality. D
REMARK 29.7. Compare 6. 9 ln v = R - 2v with 6.f = R in (28.42). In thecompact case we had g = e-f geuc, whereas in the present noncompact case we have
g = e-Inv g 5 2. So ln v in the noncom pact case plays the role of f in the compact
case.