8 2. QUASICONFORMAL MAPPINGS
2.4. Classical Regularity Theory
Recall that our definition of a quasiconformal mapping / requires that / lies
in the Sobolev space W^c (ft). Consider the radial stretching defined on the unit
disk B using
(2.12) p(t)=ta, 0 a l .
We see that
(2.13) K(z, h) = - , \Dh(z)\ = ta~\ J(z, h) = a t2a~2
and hence h G W1,P(B) for all p such that
1 - a K-\
and in no better Sobolev space. Notice that
I I II
For various reasons, this mapping was considered extremal and the above con-
clusion was conjectured to hold in complete generality. Let us discuss this a little
before revealing the complete solution.
In studying solutions to the Beltrami equation an operator, analogous to the
Hilbert transform, was introduced. This operator is now known as the Beurling-
Ahlfors transform . It is defined as a singular integral of Calderon-Zygmund
2?rz J Jc
for all functions u £
1 p oo. More precisely, the integral is understood
by means of the Cauchy principal value. We refer to Stein's book  as a general
reference for general facts concerning singular integrals. The operator S :
is bounded for all 1 p oo and is an isometry in
We denote the
p-norms of the operator S by ||5||p, so ||5||2 = 1. The characteristic property of
this operator, and the property which makes it very important to complex analysis,
is that it intertwines the d and d derivatives *.
It was Bojarski [21, 22] who first gave the elegant analytic proof of the exis-
tence and uniqueness for the Beltrami equation using this operator from which he
developed the LP theory of planar quasiconformal mappings.
Closely related to the operator S is the complex Riesz potential
(2.16) TU(z)= * " "WdCAdCz)-C(
2vr i J Jc
There are in fact two homotopy classes of first order elliptic operators in the complex plane.
These are represented by d and d. The null sets of these operators are mappings preserving
and reversing orientation respectively (holomorphic or antiholomorphic). Thus the operator S
provides a mechanism for us to move from one homotopy component to the other.