1549055259-Ubiquitous_Quasidisk__The__Gehring_

70 5. CRITERIA FOR EXTENSION

PROOF. If Bo is any disk with center z 0 and Bo c D , then

iuBo - u(zo)I :S: m(~o) l

0

lhD(z, z1) - hD(zo, zi)I dm

:::; m(~o) l

0

hD(z, zo) dm:::; 2

(5.1.3)

by Lemma 4.3.2, whence

(

~) r lu(z) - UBol dm:::; (~) r (lu(z) - u(zo)I + 2) dm:::; 4.

m o }~ m o }~

D

If v E BMO(R^2 ) and if u is the restriction of v to D, then u is in BMO(D)

with

llullBMO(D) :::; llvllBMO(R2)·

The following example shows that the converse is not always true for BMO(D) even

though this is trivially true for the class L^00 (D) in (5.1.1).

EXAMPLE 5.1.4. If u(z) = hD(z, 1) where Dis the half-strip

D={z=x+iy: O<x<oo, IYl<l},

then u is in BMO(D) but u has no extension v in BMO(R^2 ).

Our proof of this depends on the following inequality, which will also be needed

in Section 9.2.

LEMMA 5.1.5. If u is in BMO(D) and if B1, Bo are disks with B 1 C Bo CD,

then

luB, -UBol:::; ~ (1og:~!~~ +1) llullBMO(D)·

PROOF. Suppose first that m(Bo):::; em(B 1 ). Since

{ (u-UB 0 )dm+ { (u-UB 0 )dm= { (u-UB 0 )dm=0,
}Bo\B1 JB, }Bo
we obtain

luB^0 - UB^1 I= I m(~i) l, (u - UB^0 ) dml =I m(~i) lo\B, (u - UB^0 ) dml

and

1 1 r e

luBo - UB, I :S: 2 m(Bi) } Bo lu - UB 0 I dm :S: 2 llullBMO(D)

by our assumption.
Next let k be the smallest integer for which m(Bo) :::; ek m(B 1 ) and choose
disks Bj so that

and

for j = 1, 2, ... , k. Then

k k

luBo - UB, I :S: L luB; - UB;+1 I :S: -f llullBMO(D)

j=l

e ( m(Bo) )

:::; 2 log m(B1) + 1 llullBMO(D)