1549055259-Ubiquitous_Quasidisk__The__Gehring_

5.3, QUASICONFORMAL MAPPINGS 73

DEFINITION 5.2.4. A domain D is a Sobolev extension domain if for each p ,

1 ::::; p < oo, there exists a constant cp 2: 1 such that each function u in Wf (D) has

an extension v in Wf (R^2 ) such that

llvllwf(R^2 ) :S cp l!ullwf(D)·

THEOREM 5.2.5 (Jones [95]). A bounded simply connected domain Dis a qua-

sidisk if and only if it is a Sobolev extension domain.

5.3. Quasiconformal mappings

If D is a disk or half-plane, then each K-quasiconformal self-mapping f of D

can be extended by reflection to yield a K-quasiconformal self-mapping g of R

2

(Lehto-Virtanen [117]).

DEFINITION 5.3.l. D is a quasiconformal extension domain if there exists a

constant c 2: 1 such that each K-quasiconformal se lf-mapping f of D has a cK-

quasiconformal extension g to R

2
.
EXAMPLE 5.3.2. D is a disk or half-plane if and only if it is a quasiconformal
extension domain with c = l.
Our proof of this result depends on the following characterization for Jordan
domains (Erkama [36], Hag [77]).

LEMMA 5.3.3. A simply connected domain D is a Jordan domain if and only
if every conformal self-mapping of D has a homeomorphic extension to D.
PROOF. If D is a Jordan domain, then every conformal self-mapping of D
has a a homeomorphic extension to D by a well-known theorem of Caratheodory
(Pommerenke [145]). For the converse we show that 8D is lo cally connected with no
cut points, and hence a Jordan curve (Pommerenke [145]), whenever each conformal
self-mapping of D has a homeomorphic extension to D.
For the local connectivity, suppose that g is a conformal map of the unit disk
B onto D. Then there exists a point z 0 E 8B at which g has a radial limit
(Pommerenke [145]). Let
lim g(r zo) = wa.
r-+l

Next for each z 1 E 8 B let h(z ) = zi z. Then by hypothesis, the mapping
Zo
f =go h o g-^1 : D---+D

has a homeomorphic extension f* : D---+D. In particular,

lim g(r zi) = lim go h(r zo) = lim f o g(r zo) = f(wo).
r-+l r-+l r-tl
Since z 1 is an arbitrary point in 8 B , we conclude that g h as a radial limit
at each point of 8 B. Arguing again as above, we see that g has a limit at each
point of 8B. Thus g has a continuous extension to B and 8D is locally connected
(Pommerenke [145]). Finally, there exists w 0 E 8D which is not a cut point of 8D
(Pommerenke [145]). But this implies that 8D is cut point free. For let w 1 be any
point in 8D \ {w 0 } and choose z 0 ,z 1 E 8B such that g(zo) = wo and g( z1) = w 1.
As before, w1 = f*(wo),

8D \ {wi} = f(&D) \ {f(wo)} = f*(8D \ {wo})