1549055259-Ubiquitous_Quasidisk__The__Gehring_

5. CRITERIA FOR EXTENSION

EXAMPLE 5.2.1. If Dis a disk or half-plane and if u E Li(D), then u has an
extension v E Lr(R^2 ) with

However this result does not hold for all domains D. For example let D again
be the half-strip

D = { z = x + iy: 0 < x < oo, IYI < 1}

and for j > 1 and z = x + i y E D let

{

-1

Uj(z) = ~ - j

if 0 < x:::; j - 1,

if j - 1 < x:::; j + 1,

if j + 1 < x < 00.

Then ED(uj) = 4.
Suppose that Vj is an extension of Uj in Li(R^2 ) and let Cr denote the boundary
of the square with corners at (j - r, ±r) and (j + r, ±r) for 1 < r < j. Then Vj
is absolutely continuous and assumes the values -1 and 1 on C,. for almost all
r E (1,j). Thus

16:::; (1 lgrad Vj(z)lldzl)

2

:::; Sr 1 lgrad Vj(z)l

2

ldzl

Cr Cr

for almost all r E (1, j) and

ER2(vj) 2 lj ([r lgrad Vj(z)l

2

ldzl) dr 2 Cj ED(uj)

where

1. Cj = 2 log)-tOO

as j-too. Hence there exists no constant c such that each u E Li(D) has an
extension v E Li(R^2 ) with

DEFINITION 5.2.2. A domain D is an Li-extension domain if there exists a
constant c 2 1 such that each function u E Li(D) has an extension v E Li(R^2 )
with

We see then that the simply connected domains with this property are qua-
sidisks.

THEOREM 5.2.3 (Gol'dstein, Latfullin, Vodop'janov [75], [76]). A simply con-

nected domain D is a quasidisk if and only if it is an Li-extension domain.

The function u is in the Sobolev space Wf (D), 1 :::; p < oo, if

llullwf (D) = (fv iulP dm) l/p + (fv igrad ulP dm) l/p < oo.

It is natural to ask if there is an analogue of Theorem 5.2.3 for the Sobolev class.
This is, in fact, true for simply connected domains which are bounded.