CHAPTER 5Criteria for extension
Suppose F denotes a certain property of functions, for example, continuity
or integrability, and let F(D) be the family of functions defined on D with the
property F. When does each function in F(D) have an extension to R^2 which is
in F(R^2 )? As in Chapter 4, the answer depends on the nature of Fas well as the
domain D.
We consider here four different properties F for which such an extension is
possible whenever D is a disk or half-plane and then observe that the same is
true if and only if D is a quasidisk. Two of these conditions deal with real-valued
functions and the rest with the quasiconformal and bilipschitz classes.
5.1. Functions of bounded mean oscillation
We have already introduced the BMO-norm of the Jacobian J1 of a locally
quasiconformal mapping f in order to study when f is injective. We consider now
the family of functions for which this norm is finite.
Suppose that u is a locally integrable real-valued function in a domain D c R^2.
We say that u has bounded mean oscillation in D , or is in BMO(D) if
llullsMO(D) =Sup (lB ) ( lu - UB 0 ldm < 00,
Bo m O }Bowhere as in (4.3.1), the supremum is taken over all disks Bo with Bo c D and
UBo = m(~o) l o u dm.
Functions of bounded mean oscillation occur naturally in many parts of math-
ematics. They were first studied in connection with problems in elasticity (John,
Nirenberg [90], [93]) and partial differential equations (Moser [134]). They were
later found to play an important role in harmonic analysis. See Pefferman [39],
Pefferman-Stein [40], Garnett [44], and Reimann-Rychener [147].
We have that
(5.1.1) L^00 (D) c BMO(D).
It is easy to see that the inclusion is strict when D is simply connected.
LEMMA 5.1.2. If DC R^2 is simply connected, then for each z 1 ED the hyper-
bolic distance u(z) = hD(z, z1) is in BMO(D) with
llullsMO(D) :::; 4.Hence, in particular, u E BMO(D) \ L^00 (D).
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