- CLASSIFY ING THE BACKWARD POINTWISE LIMIT 101
circular averages as well as on a concentration-compactness-type result which im-
plies that v°" ~ limt__,-= v(t) has at most two zeros if it does not vani sh identically.
The classification t hen uses t he fact that an entire harmonic function on the plane
with at most linear growth must be a linear function.
9.1. An estimate from monotonicity of circular averages.
Let p E 52. We shall find it convenient to consider the conformal factors
relative to the Euclidean and cylinder metrics; see (29.4), defined via stereographic
projection u: 52 - {p} ~ JR^2. Let w ~-Inv= lnii. on JR^2 and W± ~ max{±w,O}.
Define the backward limit u°"(r, B) = limH-= u(r, B, t). We have the following
result , which is similar to Lemma 28.49.
LEMMA 29.32 (A bound for integrals of iii+ over balls). Suppose that u°"(O) <
oo, where 0 E JR^2 is th e origin. Then for any r 0 > 0 and t E (-oo, -1]
(29.99) r r
0w+(r, e , t)rdrde:::; 1rr6 ln u()()(O) + C(ro, Umin(-1)),
151 l o
where Umin(t) ~ minxE52 u(x, t) > 0.
PROOF. Define t he circular averagesW(r, t) ~ ( w(r, e, t)dB
151for r > 0. By (29.7), we have that w is superharmonic:
1 8 ( aw) 1 8
2
w
0 > -Rgu = 6.eucW = :;: or r or + r2 ae2on JR^2 - {O}. Integrating this inequality over circles yields %r(r^88 1;:'(r,t)) < 0.
Since limr').o(r^88 1;:' (r, t)) = 0, we obtain the monotonicity:^88 1;:' (r, t) < 0 for r > 0.
Therefore
( w(r,B,t)dB < lim W(r, t) = 27rw(O, t):::; 27rlnu°"(O ) < oo,
l 51 r'\,O
where t he last inequality is true because ~~ < 0. Integrating this inequality with
respect to r yields
(29.100) f r
0
w(r,B,t)rdrdB:::; 7rr6lnii.oo(O).
151 l oSince ~~ < 0 and by (29.6), we have that
4umin(-1)(29.101) u(r, e, t) ~ (1 + r2)2 fort:::; -1.
Hence, fort E (-oo,-1],
(29.102)
r ro r ro ( 4u. ( -1) )
151lo w_(r, e , t)rdrdB:::;
151
l o ln (lm~n r 2 ) 2 _ rdrdB:::; C(ro, Umin(-1)).Since iii+= w + w _ , the lemma follows from (29.100) and (29.102).^0