1549055259-Ubiquitous_Quasidisk__The__Gehring_

100 8. FIRST SERIES OF IMPLICATIONS

8.1. Quasidisks and hyperbolic segments
We begin with the following elementary observation.
LEMMA 8.1. l. Suppose that D is a disk or half-plane, that D ' is a K -quasidisk,
and that f : D-+ D' is conformal. Then f has an extension which is K^2 -quasicon-
-2
formal in R.

• 2 -2
  PROOF. By hypothesis there exists a K-quasiconformal mapping g : R -+R
  with g(D') = B. Then h =go f is a K-quasiconformal mapping of D onto B which
  extends as a homeomorphism of D onto B by Theorem 1.3.11. Let¢ and 'ljJ denote
  the reflections in oD and 8B, respectively, and set

h(z) ='l/Jogofo¢-^1 (z)

for z E D. Then h is K-quasiconformal in D , in D, and hence in R

2
by Theo-
rem 1.3.12. Set
f(z) = g-^1 o h(z)

for z ED. Then f is K^2 -quasiconformal in D , in D , and hence in R

2

. D
The next result will allow us to estimate the length of a hyperbolic segment.
The proof is based on an argument due to Jerison and Kenig [88].
LEMMA 8.1.2. Suppose that D is a K-quasidisk with oo E oD and that f :
H--+ D is a homeomorphism which is conj ormal in H with f ( oo) = oo. Then

(8.1.3)

ry

lo If' (it) I dt ::::; C1 dist(f (i y), oD)

for 0 < y < oo where c1 = c1(K).

PROOF. By Lemma 8.1.1, f has an extension which is K^2 -quasiconformal in

R

2

. By a change of variables we may assume that f(O) = 0. Fix 0 <Yo < oo and
choose a sequence {yj} so that
0 < Yi+l < Yj ::::; Yo
and
lf(iyj)I = c2j lf(iyo)I

for j = 1, 2, ... , where c2 = e^8 K

2

• Then for Yj+l ::::; y ::::; Yj,
  dist(f(iy),8D)::::; lf(iy) - f(O)I::::; c2 lf(iy1) - f(O)I = c2^1 +i lf(iyo)I
  by Theorem 1.3.4 since f (0) = 0, while

lf

'(i )I < 2 dist(f(i y), oD) < 2 c-Hl lf(i Yo)I

Y - dist(iy,oH) -^2 y

by the Koebe distortion theorem. Thus

(8.1.4) f,Y

1

lf'(iy)I dy::::; 2c2j+llf(iyo)l log-1L.

Y;+l Y1+1

Next let k denote the smallest positive integer for which c 2 ::::; 2k. Then

lf(iy1) - f(O)I::::; 2k lf(iYj+i) - f(O)I,

and hence