1549055259-Ubiquitous_Quasidisk__The__Gehring_

5 .4. BILIPSCHITZ MAPPINGS 75

THEOREM 5.4.3 (Gehring [50], Tukia [160], [162]). A bounded Jordan domain
D is a K -quasidisk if and only if it has the bilipschitz extension property with
constant c, where K and c depend only on each other.
Necessity also holds in the above theorem when D is unbounded. However
sufficiency fails in this case. For example, the half-strip

D = { z = x + i y : 0 < x < oo, IYI < 1}

has the bilipschitz extension property but D is not a quasidisk by Example 1.4.5.