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6.3. Lip"-EXTENSION DOMAINS 81

6.3. Lip 0 -extension domains

We return to Theorem 6.2.4. This result can be viewed as a consequence of

two implications. First, if (6.2.5) holds in D , then by Theorem 6.2.3

lf(zi) - f(z2)I::; comlz1 - z2I°'
a
in each open disk in D. Thus f belongs to the local Lipschitz class locLip 0 (D).
Next, the fact that Dis uniform implies that f is in Lip 0 (D) with an appropriate
norm. In general a domain D with the property that there exists a constant A=
A(D, a) such that
II! Ila ::; A II! ll~c
for all real-or complex-valued functions fin the local Lipschitz class locLip 0 (D)
is called a Lip 0 -extension domain. This class of domains can be characterized by
the following distance inequality proved in [66].
THEOREM 6.3.1 ( Gehring-Martio [66]). Suppose a E (0, l]. A domain D in
R^2 is a Lip 0 -extension domain if and only if there exists a constant M = M(D, a)
such that for all z1, z2 E D

(6.3.2) d 0 (z1, z2) =inf 1dist(z,8D)°'-^1 ds ::; Mlz1 - z2 I°'
'Y 'Y
where "( is a rectifiable curve joining z 1 and z2 in D.
Now let
Do(z1, z2) =sup lf(z1) - f(z2)I,
f
where the supremum is taken over all analytic functions f on D satisfying (6.2.7).
Then the metrics d 0 introduced in (6.3.2) and 60 are comparable. This is the
content of the next important result due to Kaufman and Wu.

THEOREM 6.3.3 (Kaufman-Wu [99]). If D is a simply connected domain in
R^2 , then

(6.3.4)
where c 1 is an absolute constant.
It turns out that the simply connected Lip 0 -extension domains for a fixed a
are exactly the Hardy-Littlewood domains of order a.

THEOREM 6.3.5 (Astala-Hag-Hag-Lappalainen [15]). A simply connected do-

main in R^2 has the Hardy-Littlewood property of order a E (0, 1] if and only if it

is a Lip 0 -extension domain.

PROOF. Assume that D has the Hardy-Littlewood property of order a. Then

and thus by (6.3.4)

c(D) 0

Do(z1, z2)::; --lz1 - z2I

a

c1c(D) 0
do(z1, z2)::; --lz1 - z2I ,
a
which shows that condition (6.3.2) is satisfied and hence that D is a Lip 0 -extension
domain.