4 1. INTRODUCTION 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 appendices. 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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