10 2. QUASICONFORMAL MAPPING S

THEOREM

2.2. Let /i be a measurable function in a domain ft c C and suppose

||/i||oo 1- 77&en £/iere is a quasiconformal mapping g : Q — C whose complex Bel-

trami coefficient is equal to fi almost everywhere. Moreover every WZo'c (fi) solution

f to the Beltrami equation is of the form

f(z) = F(g(z))

where F : g(Q.) — C is a holomorphic function.

The main results of this paper are sharp generalizations of these two funda-

mental theorems in the degenerate elliptic case. That is when ||/x||oo = 1- This

follows and extends the pioneering work of David [33]. As discussed in the intro-

duction, there is also related earlier work of Lehto on these more general problems,

see [72, 73] where a number of the necessary estimates can be found.

Let us now turn to a more general discussion.