1549055259-Ubiquitous_Quasidisk__The__Gehring_

4.3. LOCALLY QUASICONFORMAL MAPPINGS 65

for locally quasiconformal mappings. It turns out t hat the BMO (bounded mean

oscillation) norm of the Jacobian J1 of a locally quasiconformal mapping f is a

natural counterpart for each of the above quantities.

Suppose that u is a function locally integra ble in a domain D C R^2. Then t he

BMO-norm of u in Dis given by

(4.3.1) llullsMO(D) =sup (lB) ( lu - UB 0 l dm

Bo m 0 }Bo

where the supremum is taken over all disks Bo with Bo c D and

UBo = m(~o) l o udm.
The following observation suggests how this norm is related to the hyperbolic
metric.

LEMMA 4.3.2. If Bo is a disk in D with center z 0 , then

m(~o) l

0

hD(z, zo) dm '.'::: 2.

PROOF. The left-hand side of the above inequ ality is invariant with respect to
similarity mappings. Hence we may assume that B 0 is the unit disk B. Then

L hD(z, O)dm '.'::: L hB(z, O)dm = L log(~~::: )dm

= fo

2

~ (fo

1

logC ~ ~) tdt) dB

[

t^2 - 1 (l+t)]t=l

= 27r t + --log --=- = 2m(B).

2 1 t t=O

0

With Lemma 4.3.2 we are able to derive the following useful estimate for the
BMO norm of a harmonic function.

LEMMA 4.3.3. If u is harmonic in D c R^2 , th en

~ llullsMO(D) '.':::sup !grad u(z)I PD
1
(z) '.'::: 6llullsMO(D)
2 zED
where PD is the density of the hyperbolic metric hD.
PROOF. Choose a disk Bo = B (z 0 , d) with Bo C D. Then since u is harmonic
in D , u( zo) = UB 0 and it suffices to prove that

(4.3.4) !grad u(zo)I PD(zo)-

1

'.'::: m(~o) l

0

lu(z) - u(zo)I dm

and

( 4.3.5) _ (l ) r /u(z) - u(zo)ldm::::; 2 sup lgrad u(z)I PD(z)-^1.

m Bo }Bo zED

By performing a change of variable we may assume that zo = 0.

For (4.3.4) fix 0 < r < d and let

f(z ) = J_ { 2 ~ re•;+ z u(rei&) dB.

27r } 0 re' - z