1549055259-Ubiquitous_Quasidisk__The__Gehring_

9.4. UNIFORM DOMAINS ARE RIGID 129

LEMMA 9.4.10. If Eis a convex set in R^2 and if f: E-+R^2 is locally L-lipschitz

in E, then f is L-lipschitz in E.

PROOF. Suppose that zo, wo EE. Then [z 0 , w 0 ] is a compact subset of E and

there exists a o > 0 such that

(9.4.11) lf(z) -f(w)I:::; Li z - wl

whenever z, w E E and lz - wl < o. Next choose an ordered sequence of points

z1, z2, ... , Zn+l = wo in [zo, wo] such that

for j = 0, 1, ... , n. Then

n

lf(zo) -f(wo)I:::; L lf(zj) -f(zj+i)I

by (9.4.11).

j=O

n

:::; LL lzj - ZJ+1I =Liza - wol

j=O

D

REMARK 9.4.12. The conclusion in Lemma 9.4.10 also follows with the hy-

pothesis that f is locally L-lipschitz replaced by the seemingly weaker assumption

that

limsup lf(z) - f(w)I < L

z-+w lz - wl -

for each w EE.

See John [91] and Nevanlinna [138].

LEMMA 9.4.13 (John [91]). If f : B(zo, r)-+R^2 is locally L-bilipschitz in B(z 0 , r),

then f is L-bilipschitz in B(z 0 , r / L^2 ).

PROOF. We may assume without loss of generality that f(zo) = zo = 0 and

r = 1. Then Lemma 9.4.10 implies that f is L-lipschitz in B and hence that

f(B(o,L-^2 )) c B(o,L-^1 ).

To complete the proof, we shall define a continuous function g on B(O, L-^1 ) such

that

(9.4.14)

(9.4.15)

f(g(w)) = w

g(f(z)) = z

for

for

wE B(O,L-^1 ),

z E B(O, L-^2 ),

for if g is continuous, (9.4.14) will imply that

~ < lf(g(w1)) - f(g(w2))I = lw1 - w2I

L - lg(w1) -g(w2)I lg(w1) - g(w2)I

locally in B(O, L -^1 ) and hence that g is L-lipschitz in B(O, L -l) by a second appli-
cation of Lemma 9.4.10, and with (9.4.15) we will conclude

~ < lf(z1) -f(z2)I = lf(z1) - f(z2)I

L - lg(f(z1)) -g(f(z2))I lz1 - z2I

when z 1 , z 2 E B(O, L -^2 ) and hence that f is L-bilipschitz in B(O, L-^2 ).