1549055259-Ubiquitous_Quasidisk__The__Gehring_

114 8. FIRST SERIES OF IMPLICATIONS

for j = 1, 2, ... , n. If, in particular, h('Y) is the segment joining h(zi) and h(z2),
then
n n
lh(z1) - h(z2)I::; L lh(wj) - h(wj+i)I::; ML lh(wj) - h(wj+1)I
j=l j=l

Similarly

n n

lh(zi) - h( z2)I::; L lh(wj) - h(wj+1)I::; ML lh(wj) - h(wj+1)I

j=l j=l

= Mlh(z1) - h(z2)I

if h(r(!')) is a segment where r(z) = z. D

THEOREM 2.1.8 (Ahlfors [5]). Suppose thatoo E 8D. ThenD is aK-quasidisk

if and only if it admits a Euclidean L-bilipschitz reflection, where K and L depend

only on each other.

PROOF. If f is a mapping which is L-bilipschitz with respect to the Euclidean

metric, then f is L^2 -quasiconformal by Definition 1.1.3 and D is a K = K(L)-

quasidisk by Theorem 2.1.4.

Conversely if D is a K-quasidisk, then D = h(H) where h : R

2

---+R

2

is K-

quasiconformal and

f(z) =ho r o h-^1 (z), r(z ) = z,

is a hyperbolic L-bilipschitz reflection in 8D where L = L(K). Ifw 1 ,w2 ED and

if Zj = h-^1 (wj), then f(wj) = h(zj) and we obtain

_!_ < lf(w1) - f(w2)I ::; M

M - lw1 -w2I

from Lemma 8.7.3. D

8.8. Quasidisks and decomposability

Recall that a domain D is K' -quasiconformally decomposable if for each z 1 , z 2 E

D there exists a K' -quasi disk D' such that

(8.8.1) z1, z2 E D' C D.

We conclude this chapter by showing that quasiconformal decomposability charac-

terizes the class of quasidisks.

THEOREM 2.5.3. A simply connected domain D is a K-quasidisk if and only

if it is K' -quasiconformally decomposable, where K and K' depend only on each

other.

PROOF. A K-quasidisk is clearly K'-quasiconformally decomposable with K' =

K.

For the converse we may assume by means of a preliminary Mobius transfor~

mation that D C R^2. Then for each z 1 , z 2 E D, there exists a K' -quasidisk and a