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10.2. BMO-EXTENSION AND THE HYPERBOLIC METRIC

PROOF. Choose z 1 , z 2 ED with

ri = dist(z1, aD) S dist(z 2 , aD) = r 2

and let t = fo(z 1 , z2). If 1 St< oo, then

hD(z1,z2) s (a+DJD(z1,z2) s (a+b)fo(z1,z2).

If 0 < t < 1, then

(

lz1 - z2I + 1 )

2

Set and s = lz1 - z2I S et/2_1 < l.

r2 r2

Thus z1, z2 ED'= B(z2, r2) CD and

hD(z1, z2) S hD^1 (z1, z2) =log C ~;) St+ t^2 < 2fo(z1, z2).

Inequality (10.2.2) then follows from what was proved above.

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THEOREM 10 .2.3 (Jones [94]). If D has the BMO-extension property with con-

stant a, then

(10.2.4)

for z1, z2 E D where c = c( a).

PROOF. Fix z 1 , z2 E D and let

u(z) = hD(z, z1)

for z E D. Then by Lemma 5.1.2, u is in BMO(D) with

llullsMO(D) s 4.

Next, by hypothesis, u has an extension v in BMO(R^2 ) with

(10.2.5) llvllsMO(R^2 ) Sa llullsMO(D) S 4a.

For j 1 ,2 let Bj = B(zj,rj) where ri = dist(zj,8D). By relabeling if

necessary we may assume that ri S r2. Next let

Bo= B(z2,ro) where ro = lz1 - z2I +r1.

Then

lz - z2I < dist(z2, aD) s lz1 - z2I + dist(z1, aD) = ro

if z E B 2 , and Bj C Bo for j = 1, 2. Hence

lvBj -VBal S 2ae (log:~!;~+ 1) S 4aelog Czi ~ z

2

1+1) + 2ae

by Lemma 5.1.5 and we obtain

luB, - UB 2 I = lvB, -VB 21 S bfo(z1, z2) + b

where b = 4 a e. Finally,