1549055259-Ubiquitous_Quasidisk__The__Gehring_

6.4. HARMONIC DOUBLING CONDITION 85

Next we will use the doubling condition in D*. For this we choose n :::; bas above

so that diam('Y(wn, z3)) < diam('Y(z1, Wn)) and w(r(wn, z3)) :::; bw(r(z 1 , wn)).

Hence

(6.4.9)

w(r(z2, z3)) = w(r(z2, Wn)) + w(r( Wn, Z3))

:::; (1 + b) w(r(z1, Wn)).

Moreover, fork= 0, 1, ... , n - 1 we have that

diam(!'( wk, Wk+i)) = diam('Y(z 1 , wk))

and

From this we see that

w*(r(z1, Wk+1)) = w*(r(z1, wk))+ w*(r(wk, Wk+1))

:::; (1 + b)w*(r(z1,wk)),

for k = 0, 1, ... , n - l. Combining (6.4.9) with these n inequ alities we get

w*(r(z2, z3)):::; (b + l)n+l w*('Y(z1, z2))

:::; (b + l)b+^1 w*(r(z1, z2)).

All in all we have that

1

bw*(r1):::; w*(r2):::; (b+ l)b+lw*(r 1 )

and D is quasisymmetric with c = (b + l)b+^1.
Conversely, assume that D is a bounded quasidisk and let ¢: D ---+ B b e a
conformal mapping. Then ¢ has an extension to a quasiconformal self-mapping of
-2
R which we also denote by¢. See Corollary 2.1.5. By composing¢ with a Mi:ibius
transformation, if necessary, we may assume that ¢( oo) = oo. Let 11 and 12 be
adjacent arcs in 8D with diam(r 1 ) = diam(r 2 ) and let the pairs z 1 , z 2 and z 2 , z 3
b e their endpoints, respectively. By Ahlfors' three-point condition (Theorem 2.2.5)
we have that
lz3 - z2I :::; diam('Y2) = diam('Y1) :::; alz2 - z1I,

where a= a(D). By the distortion result in Corollary 1.3.7 we obtain

l¢(z3) - ¢(z2)I :S: cl¢(z2) - ¢(z1)I,

c = c(a). In the following we assume without loss of generality that the arc ¢(1' 2 )

is the larger arc. If this arc subtends an a ngle of less than 7f at the origin, then

7f

w(O, ¢(1'2); B) :::; c

2

w(O, ¢(1'1); B).

In the case when ¢(1' 2 ) subtends an angle larger than 7f we look at the subarc of
¢(1'2) from ¢(z2) to ¢(z3) of length 7f and apply Ahlfors' three-point condition to
/~ = 1(z2, z3) and 1i = 1(zi, z2) of t he same diameter and proceed as before. In
this case we have that

w(O, ¢(1'2); B) :::; 7rcw(O, ¢(1'1); B),

and we conclude that D is doubling. In the same way we prove that D* is doubling.
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