1549055259-Ubiquitous_Quasidisk__The__Gehring_

6.4. HARMONIC DOUBLING CONDITION 83

Finally, we have a characterization of quasidisks in terms of Lipa-extension

domains parallel to Theorem 6.2.10.

THEOREM 6.3.10 (Gehring-Martio [66]). Suppose D is a simply connected do-

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

is a Lipa-extension domain and D* is a Lip.a-extension domain with a, f3 E (0, 1].

This result, as well as Theorem 6.2.10, follows from the fact that the domains

D and D* possess property 1° in Definition 2.4. l according to Theorem 6.3. 7 and

Theorem 6.1.2 applies.

6.4. Harmonic doubling condition

In Sections 3.8 and 3.9 we saw how quasidisks can be described by conditions in-

volving harmonic measures. Here we will give yet another characterization in terms

of harmonic measure, but this time the condition has to be satisfied simultaneously

in D and D*.

DEFINITION 6.4.l. Let D C R

2

be a Jordan domain with bounded boundary

8D. We say that D satisfies a harmonic doubling condition, or simply that D is

doubling, if there exists a point zo E D and a constant b 2". 1 such that if 11 and (^12)
are adjacent arcs in 8D with
diam(r 2 ):::; diam(r1),
then
w(zo, 12; D) :S bw(zo, 11; D).
The inequality for the diameters may be replaced by an equality in the above
definition.
The doubling condition is clearly satisfied in B and B , but not in H since
(3.7.2) shows that in this case boundary arcs of large Euclidean diameters need not
carry much harmonic measure. Hence it is necessary to restrict our attention to
domains whose boundaries lie entirely in R^2.
REMARK 6.4.2. A bounded Jordan domain Dis a disk if and only if D and D
both satisfy a harmonic doubling condition with constant b = l.
PROOF. To establish sufficiency, suppose that D and D satisfy a harmonic
doubling condition for b = 1 and the points z 0 E D, z 0 E D, and let 11 and
12 be consecutive boundary arcs with diam(r 1 ) = diam(r2). Then w(zo,/1;D) =
w(zo, 12; D) and w(z 0 , 11 ; D) = w(z 0 , 12 ; D). It is not difficult to see that
(6.4.3) w(zo, 11; D) = w(zo, 12; D) implies diam(r1) = diam(r2),
and the same conclusion holds in D as well.
Let ¢: B ---+ D and 'ljJ: B ---+ D be conformal maps with ¢(0) = z 0 and
'l/J( oo) = z 0. Let Ii, h C S^1 = f)B be consecutive arcs with length( Ii) = length(I2)
or, equivalently, w(O, Ii; B) = w(O, h B). Then by conformal invariance w(zo, ¢(Ii);
D) = w(zo, ¢(h); D), and (6.4.3) implies that diam(¢(li)) = diam(¢(h)). The
doubling condition yields
w(z 0 , ¢(Ii); D) = w(z 0 , ¢(!2); D*),
so that by conformal invariance once more we obtain
w(oo, f(I 1 ); B) = w(oo, f(I2); B),