2.4. CLASSICAL REGULARITY THEORY

9

Green's formula gives dTuj = LU and dTuj = SLU for smooth uu with compact support.

The density of smooth functions in

Lp

and the

Lp

boundedness of these operators

implies that all the above formulas hold whenever

LU

G

LP(C).

Now a solution / to the complex Beltrami equation with compactly supported

// can be found in the form f(z) = z +

TLU(Z),

where

(2.17) LU = fi + fiSuj almost everywhere

We shall call / the principal solution.

The integral equation at (2.17) is uniquely solved by the Neumann iteration

procedure in

LP(C)

if

(2-18) |MU|S||

P

1

The fact that it is the invertibility of the Beltrami operator I — fiS :

LP(C)

—

LP(C)

which determines the Lp theory of solutions to the Beltrami equation as was first

observed by Bojarski, [21, 22]. The above representation of the solution / to the

Beltrami equation shows that / G z + WliP(C) for all p such that ||A*||oo||S'||p 1-

Since \\S\\2 = 1 and ||/i||oo 1 there is always such a p 2. This implies a higher

degree of regularity for the principal solution than the initial assumption that it

lies in Wj0'c (C). The factorization theorem then shows all W^c (C) solutions enjoy

this higher degree of regularity, see [3, 75, 22].

This is all clearly explained in Lehto and Virtanen's book [75] where they an-

ticipate the optimal results which were finally obtained by Astala [6] in spectacular

fashion in 1993 using ideas from holomorphic dynamics. Together with previously

known and more elementary results, the "Area Distortion Theorem" of Astala can

be stated in the following way:

THEOREM

2.1. Let

/J,

be a measurable function defined in ft with ||/i||oo

=

fcl.

Let f be any solution to the Beltrami equation with f G Wt^(Q), q 1 + k. Then

f G Wj0'^(£i) for all p 1 -f ^. This theorem is sharp; there may be solutions in

Wj0'c+ (ft) not in any higher Sobolev space, and there may be solutions in W^c (ft)

notinWl+l/k(ti).

Notice the indices p and q in the above result form a Holder conjugate pair.

See [10] for a more general discussion and further results in this direction.

Indeed Theorem 2.1 would follow from the methods outlined above if the con-

jectural values ||SUp = p — 1 for p 2 were to be proven. This perhaps is one of

the most important outstanding problem in the planar

theory2.

It is not too difficult to go from Bojarski's representation formula to the exis-

tence theorem. The existence theorem for quasiconformal mappings, more recently

called the "measurable Riemann mapping theorem", is one of the most fundamen-

tal results in the theory and has come to play a central role in modern complex

analysis, [87, 19].

There have been important partial results towards the solution of this problem obtained by

Volberg and Nazarov [110] improving earlier estimates of Banuelos and Wang [14] and ourselves

[60]. The current best estimate is ||5||

p

2(p - 1) for p 2.