1549055259-Ubiquitous_Quasidisk__The__Gehring_

2.1. REFLECTION 21

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A domain DC R is a Jordan domain if and only if it admits a reflection fin

its boundary. What else can we say about D if we know more about the reflection

f? When, for example, can we conclude that D is a quasidisk?

One obvious situation is as follows.

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THEOREM 2.1.4 (Ki.ihnau [108]). A domain DC R is a K-quasidisk if and

only if it admits a K -quasiconformal reflection in DD.

PROOF. If D admits a K-quasiconformal reflection fin DD, then Dis a Jordan

domain by Theorem 2.1.2. Hence there exists a homeomorphism h which maps D

onto the upper h alf-pla ne H and is conformal in D. In this case

g(z ) = {h(z)

rohof-^1 (z) ifzED*,

if z E D ,

where r(z) = z, defines a K-quasiconformal se lf-mapping of R

2
with g(D) = H
and D is a K-quasidisk.
If D is a K-quasidisk, then there exists a K-quasiconformal self-mapping g of
-2
R which maps D onto the upper half-plane Hand
f(z) = g-^1 or o g(z)
defines a K^2 -quasiconformal reflection in DD.
A more detailed argument based on Theorem 1.1.11 shows that the mapping f
can b e replaced by a K-quasiconformal mapping f* and hence that D is actually
a K-quasidisk. See Kiihnau [108]. 0
COROLLARY 2.1.5. A quasiconformal mapping between two quasidisks can be
extended to a quasiconformal se lf-mapping of R

2
.
Another far less obvious situation concerns the case when the reflection f is
bilipschitz with respect to the Euclidean metric.

DEFINITION 2.1.6. A mapping f : E-+ E' is an L-bilipschitz mapping, or more
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precisely a E1iclidean L-bilipschitz mapping, of E C R if
1
L lz1 - z2I :S lf(zi) - f(z2)I :SL lz1 - z2I

for z 1 , z2 EE\ {oo} and f(oo) = oo if oo EE.

If Dis a h alf-plane, then there exists a 1-bilipschitz mapping of R
2
that maps D
onto its exterior D* and is the identity on DD. More generally a sector D of a ngle
a admits an L-bilipschitz reflection where L is bounded by constants depending
only on a [59]. The optimal bilipschitz reflection in a sector has been determined
by J. Miller [130].

THEOREM 2.1.7 (Miller [130]). If D = S(a), 0 :S a :S -rr, then DD admits an

optimal L-bilipschitz reflection f with L = cot B, where B is the unique angle such

that 0 :Se< a/2, e :S -rr/4, and ¢(8) = 1, where

-rr + 2t - a 2

¢(t) = a_ 2 t tan t.

Moreover this reflection is unique.

If -rr :S a < 2-rr, then the unique optimal reflection is given by f, where f -^1 is

the above optimal reflection for S(2-rr - a).