1549055259-Ubiquitous_Quasidisk__The__Gehring_

4.4. JACOBIAN OF A CONFORMAL MAPPING 67

The following quasiconformal analogue of Theorems 4.1.4 and 4.2.3 shows that

the implication in Theorem 4.3.7 can also be reversed for locally K-quasiconformal

mappings in disks and half-planes provided K is not too large.

THEOREM 4.3.8 (Astala-Gehring [14]). Suppose that D is a disk or half-plane.

Then for each 1 :::; K < 2 there exists a constant c = c(K) > 0 with the following

property. If f is locally K-quasiconformal in D with f(D) c R^2 and

II log JJ llBMO(D) :::; C,

then f is injective in D. The constant 2 is sharp; i.e., no such constant c exists if

K ~ 2.

Finally we see that the class of domains D for which such a converse exists

coincides with the family of quasidisks.

THEOREM 4.3.9 (Astala-Gehring [14]). A simply connected domain D c R^2 is

a quasidisk if and only if for some K > 1 there exists a constant c > 0 such that f

is injective whenever f is locally K -quasiconformal in D with f(D) C R^2 and

lllogJJllBMO(D):::; C.

4.4. Jacobian of a conformal mapping

If f is conformal in DC R^2 with f(D) C R^2 , then the function
u =log J 1
is harmonic in D. Next Theorem 4.3.7 implies that u also has finite EMO-norm
in D. Hence it is natural to ask under wha t circumstances these two properties
characterize the Jacobian of a conformal mapping.
The answer yields still another description of a quasidisk.
THEOREM 4.4.1 (Astala-Gehring [14]). A simply connected domain D C R^2
is a quasidisk if and only if there exists a constant c > 0 such that each function u
harmonic in D with
llullBMO(D) :::; C
can be written in the farm
( 4.4.2) u = logJJ
where f is conformal in D with f(D) c R^2.
PROOF. Necessity follows from Theorem 4.3.9 above. For sufficiency suppose
there exists a constant c > 0 such that (4.4.2) holds whenever u is harmonic with
llullBMO(D) :::; c. Next choose g analytic in D with g'(z ) =I- 0 and

I

g" ( z) I - 1 < c

g'(z) PD(z) _ 4

in D and let u =log J 9. Then (4 .4.2) holds where f is conformal in D and

h(z ) = g'(z )

f'(z )

is analytic with
2log lh(z)I = J 9 (z ) - J1(z ) = 0