1549055259-Ubiquitous_Quasidisk__The__Gehring_

5.2. SOBOLEV AND FINITE ENERGY FUNCTIONS 71

by what was proved above and the choice of k. D

We turn now to a proof of the result in Example 5.1.4. Suppose that u has an
extension v in EMO(R^2 ), choose n > 5 so that

1 n-5

llvllBMO(R^2 ) < ~ 2 log n + 1'

and let Bj = B(zj, 1) and Bo= B(n, n) where z 1 = 1 and z 2 = 2n - 1. Then

lvB; - VB 0 I ::; ~ (log:~~~~ + 1) llvllBMO(R2)

for j = 1, 2 by Lemma 5.1.5, whence

(5.1.6) luB 1 - uB 2 I = lvB 1 - VB 21 < n - 5.

Next if 'Y is a hyperbolic segment joining z 1 and z2 in D, then

1

length("f)

lu(zi) - u(z2)I = ho(z1, z2) = 'Y po(z) ds 2".

2

2". n-1

by inequality (3.2.1) and

JuB 1 - uB 2 J 2". Ju(z1) - u(z2)J - Ju(z1) - UB 1 J - Ju(z2) - UB 2 l 2". n- 5

by (5.1.3), which contradicts (5.1.6).

The domain D in Example 5.1.4 is not a quasidisk by Example 1.4.5. On the
other hand, functions in EMO(D) have extensions to EMO(R^2 ) whenever D is a
disk or half-plane.

THEOREM 5.1.7 (Reimann-Rychener [147]). If Dis a disk or half-plane and if
u E EMO(D), then u has an extension v E EMO(R^2 ) with

llvllBMO(R2) ::; c JluiiBMO(D)

where c 2". 1 is an absolute constant.

DEFINITION 5.1.8. A domain D is a EMO-extension domain if there exists a
constant c 2". 1 such that each u E EMO(D) has an extension v E EMO(R^2 ) with

JlvllBMO(R2) ::; c JluliBMO(D)'
This extension property then yields another characterization for the class of
quasi disks.

THEOREM 5.1.9 (Jones [94]). A simply connected domain D C R^2 is a K-
quasidisk if and only if it is a EMO-extension domain with constant c, where K
and c depend only on each other.

5.2. Sobolev and finite energy functions
We assume throughout this section that u is locally integrable and ACL ( abso-
lutely continuous on lines) in a domain D C R^2 ; see Definition 1.1.7. We say that
u has finite Dirichlet energy or is in Li(D) if

Eo(u) = l Jgrad uJ^2 dm < oo.

When do such functions u have an extension v in Li(R^2 )?