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.
Previous Page Next Page