1549055259-Ubiquitous_Quasidisk__The__Gehring_

138 10. THIRD SERIES OF IMPLICATIONS

10.1. Quasidisks and BMO-extension

We shall need the following result due to H. M. Reimann concerning the invari-
ance of functions with bounded mean oscillation under quasiconformal mappings.

LEMMA 10.l.l. Suppose that D and D' are domains in R^2 and that f: D-+ D'

is K-quasiconformal. If vis in BMO(D'), then u = v of is in BMO(D) and

llullBMO(D) :::; a llvllBMO(D') )
where a= a(K).
The proof is based on integrability properties of quasiconformal mappings. See,
for example, Theorem 2 in Reimann [146] or Theorem V .C.3 in Reimann-Rychener
[147].
THEOREM 10.l.2 (Jones [94]). If D is a K-quasidisk, then D is a BMO-
extension domain with constant c = c(K).

PROOF. By hypothesis there is a K-quasiconformal self-mapping f of R
2
which
maps D onto a disk or half-plane D'. By composing f with a Mo bi us transformation
we may assume that f(oo) = oo.
Suppose now that u is in BMO(D) and let u' = uo f-^1. Then by Lemma 10.1.1,
u' is in BMO(D') and

llu'llBMO(D') :::; a llullBMO(D)

where a= a(K). Next by Theorem 5.1.7, u' has an extension v' in BMO(R^2 ) with

llv'llBMO(R^2 ) :::; b llu'llBMO(D'))

where b is an absolute constant. Then again by Lemma 10.1.1, v = v' of is in
BMO(R^2 ) with
llvllBMO(R^2 ) :::; a llv'llBMO(R^2 ).
Thus vis the desired EMO-extension of u and

llvllBMO(R2) :::; c llullBMO(D)'

where c = a(K)^2 b. D

10.2. BMO-extension and the hyperbolic metric

We show next that if Dis a EMO-extension domain, then the hyperbolic metric

hD(z1, z2) in Dis bounded above by a constant times the distance-ratio metric

fo(z^1 ,^22 ) =log ( di~:(z~,~1) +^1 ) ( di~:(z~,~1) +^1 ) ·

We begin with the following preliminary result.

LEMMA 10.2.1 (Gehring-Hag [56]). Suppose that D is simply connected and

that there exist constants a > 0 and b > 0 such that

hD(z1, z2) :::; afo(z1, z2) + b

for z1, z2 ED. Then

(10.2.2)

for z 1, z 2 ED where c =max( a+ b, 2).