1549055259-Ubiquitous_Quasidisk__The__Gehring_

80 6. TWO-SIDED CRITERIA

FIGURE 6.2

EXAMPLE 6.2.9. Define a Jordan domain D as

00

where 6.i is the interior of an equilateral triangle with side length 2-i. The polar

angle ei for the center and the vertex of 6.i closest to the origin is 3n(l - 2-i)/2

and the distance from this vertex to the origin is ri = 1 - 4-i /2. Then D h as the

Hardy-Littlewood property but is not a quasidisk since it violates the three-point

condition in Section 2.2.

To see that D has the Hardy-Littlewood property, let f be an analytic function

satisfying (6.2.5) in D and choose any z 1 , z 2 in D. Now Band each 6.i are quasidisks

and hence uniform domains by Theorem 3.4.5. Thus if z 1 and z2 both lie in B or

in some 6.i, Theorem 6.2.4 shows that f lies in Lip 0 ,(B) or Lipa(6.i) · Next if

z 1 E 6.i \ B and z 2 E B \ 6.i, there is a point w E B n 6.i. By uniformity there

are curves 'Yi and ')' 2 joining z 1 and z 2 to w in 6.i and B , respectively, satisfying

(3.5.2) and (3.5.3). By generalizing the argument in the proof of Theorem 6.2.4 it

is easy to see that f lies in the right Lipschitz class. The last case when z1 E 6.i \ B

and z 2 E 6.j \ B , i of. j, follows similarly by joining z 1 and z2 to w 1 E 6.i n B and

w 2 E 6.j n B, respectively, and then joining w 1 and w 2 by a third curve in B.

We have, however, the following characterization of quasidisks in terms of the

Hardy-Littlewood propety.

THEOREM 6.2.10 (Gehring-Martio [64]). Suppose that Dis a simply connected

domain in R^2 with oo E fJD and D * a domain. Then D is a quasidisk if and only

if D and D* have the Hardy-Littlewood property.

This result will follow immediately from Theorem 6.1.2 if we can establish that

the Hardy-Littlewood property of some order implies property 1° in Definition 2.4.1.

In the original paper [64] this was done via Lemma 9.3.1. We will indicate another

route in the next section which sheds some light on the Hardy-Littlewood property

and related notions.