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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 a and hence h G W1,P(B) for all p such that 2 2K 1 P 1 - a K-\ and in no better Sobolev space. Notice that I II l ~ a 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 [5]. It is defined as a singular integral of Calderon-Zygmund type, (2.14) SLJ(Z) -If 2?rz J Jc u (()d(Ad( for all functions u £ LP(C), 1 p oo. More precisely, the integral is understood by means of the Cauchy principal value. We refer to Stein's book [102] as a general reference for general facts concerning singular integrals. The operator S : LP(C) — LP(C) is bounded for all 1 p oo and is an isometry in L2(C). 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( -II 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.