greatly refined these results and extended their utility by proving existence and
uniquess in the case that the exponential of the distortion function is Lp-integrable
(he used a slightly different but equivalent condition). Indeed David proved more
than he stated and achieved W1,2-regularity for sufficiently large p, quite close
to the results of our §11. He also established the important modulus of continuity
estimates. David's methods were entirely analytic and in spirit close to the methods
we employ here. Finally Brakalova and Jenkins (1998) established existence in the
case of subexponentially integrable distortion. Their methods were geometric using
the modulus of curve families (and so modulus of continuity arguments). Little
analytic regularity was obtained. Their existence results can also be improved
using our factorisation trick in §§12 & 13.
In this monograph we synthesize and extend these latter two approaches to
studying the degerate Beltrami equation employing analytic techniques to discuss
extistence, regularity and uniqueness in the degenerate setting obtaining results
which are very close to optimal. Our approach uses the substantial advances made
in Harmonic analysis in recent times as well as important new developments con-
cerning properties of Jacobians. A discussion of a number of these important,
but quite technical, properties of Jacobian determinants has been relegated to the
While being fully aware of the well-known aphorism of Yogi Berra "making
predictions is hard, especially about the future" we hope, and indeed expect, that
the developments in the theory of the planar Beltrami equation as presented here
will have substantial applications elsewhere. We discuss a couple of possibilities in
§17. Some problems and potential future directions are discussed in [61].
Finally we would like to thank the anonymous referee who carefully read the
paper and made a number of suggestions which improved it. In particular the
referee pointed out Lehto's papers, observed the stronger results David obtained
and saw how Brakalova and Jenkins results could be improved to what we obtain.
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