1549055259-Ubiquitous_Quasidisk__The__Gehring_

112 8. FIRST SERIES OF IMPLICATIONS

8.7. Reflections and quasidisks
Theorem 8 .6. 7 completes a circle of implications to show the equivalence of five
different characterizations of a quasidisk. We conclude this chapter by establishing
three additional descriptions for this class of domains. The first two are concerned
with reflections.
We will need the follwing lemma.

LEMMA 8.7.l. Suppose that D and D ' are simply connected domains in R

2

and

th at f: D-+D' is a homeomorphism such that

1

(8.7.2) L hD(z1, z2) :S:: hD^1 (f(z1), f(z2)) :S:: L hD(z1, z2)

for z1, z2 E D where L < oo. Then f is an L^2 -quasiconformal mapping.

PROOF. For z 0 ED\ {oo, f-^1 (00)} and l > 1 we can choose 15 > 0 so that

--PD(zo) < hD(z, zo) < l PD ( ) zo

l - Jz - zol -

and

PD^1 (f(zo)) < hD^1 (f(z), f(zo)) < l i(f( ))

l - Jf(z) - f(zo)I - PD zo

when 0 < Jz - z 0 J < 15 , where PD and PD' denote the hyperbolic densities in D and

D'. Then (8. 7.2) implies that

Jf(z) -f(zo)J < l2 hD^1 (f(z),f(zo)) PD(zo) < Ll2 PD(zo)

Jz - zol - PD^1 (f(zo)) hD(z, zo) - PD^1 (f(zo))'

Jf(z) -f(zo)I > 2_ hD^1 (f(z),f(zo)) PD(zo) > _1_ PD(zo)

Jz - zol - [2 PD^1 (f(zo)) hD(z,zo) - Ll^2 PD^1 (f(zo))

for 0 < Jz - zol < 15 and we obtain

1 pD(zo) <

1

.. f Jf(z ) - f(zo)I
Imm
L PD^1 (f(zo)) - lz-zol-+O Jz - zoJ

< 1

. lf(z) - f(zo)I < L PD(zo)
Im sup.

• lz-zol-+O Jz - zoJ - PD^1 (f(zo))
  In particular,

H ( ) 1

. SUPlz-zol=r JJ(z) - f(zo)J < L2
f zo = Im sup. _
r-+0 mflz-zol=r Jf(z ) - f(zo)J
and f is L^2 -quasiconformal by Definition 1.1.3. D
This lemma together with Theorem 2.1.4 allows us to characterize the domains
which admit a hyperbolic bilipschitz reflection.

THEOREM 2. l.11. A domain D is a K -quasidisk if and only if it admits a
hyperbolic L-bilipschitz reflection, where K and L depend only on each other.
PROOF. Suppose that D is a K-quasidisk. Then the chain of implications
established in Sections 8.1 through 8.6 and Theorem 8.6.7 imply that D admits
a hyperbolic L-bilipschitz reflection where L = L(K). Conversely if D admits a
hyperbolic L-bilipschitz reflection, Theorem 2.1.4 and Lemm a 8. 7 .1 imply that D
is a K-quasidisk where K = L. D