## The Beltrami Equation (Google eBook) |

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### Contents

5 | |

Partial Differential Equations | 11 |

Mappings of Finite Distortion | 17 |

Hardy Spaces and BMO | 27 |

The Principal Solution | 33 |

Solutions for Integrable Distortion | 39 |

Some Technical Results | 81 |

89 | |

### Common terms and phrases

analytic apply Beltrami coefficient Beltrami equation Cantor set classical compactly supported complex conformal mappings constant continuity estimates defined definition denote Department of Mathematics DF(z domain elliptic example exponent exponentially integrable exponentially integrable distortion fact factorization theorem fi(z finite distortion formula Gehring geometric Hardy spaces Hausdorff Hausdorff dimension Hence holomorphic motion holomorphically varying family homeomorphic solution homeomorphism implies Jacobian determinant Lebesgue Lehto lemma limit linear locally integrable locally uniformly log(e LP(C mappings of finite Math measurable function modulus of continuity non-linear nonconstant norm orientation preserving mapping Orlicz function Orlicz spaces Orlicz-Sobolev spaces paper PDEs planar plane pointwise proof properties quasiconformal mappings quasiregular radial stretching reader regularity relatively compact Riemann surfaces satisfies sequence shows singular Sobolev class Sobolev space solution h Stoilow Factorization subexponentially Theorem 7.2 uniform bounds unique principal solution unit disk weakly monotone

### Popular passages

Page 9 - It is not too difficult to go from the representation formula to the existence theorem. The existence theorem for quasiconformal mappings, more recently called the "measurable Riemann mapping theorem...

Page 89 - Sharp inequalities for martingales with applications to the Beurling-Ahlfors and Riesz transforms, Duke Math. J. 80 (1995) 575-600.

Page 6 - Thus the generalisation to Sobolev spaces is necessary if one is to solve extremal problems. We then find the limit of a bounded sequence of quasiconformal mappings is either quasiconformal or constant. The equivalence between the geometric definition and the analytic definition was shown by F. W.

Page ix - The first author was supported in part by grants from the US National Science Foundation. The...

Page 89 - K. Astala, Area distortion of quasiconformal mappings, Acta Math., 173, (1994), 37-60.

Page 91 - T. Iwaniec and V. Sverak, On mappings with integrable dilatation, Proc. Amer. Math. Soc., 118, (1993), 181-188.

Page 89 - LV Ahlfors, Lectures on quasiconformal mappings, Van Nostrand, Princeton 1966; Reprinted by Wadsworth Inc. Belmont, (1987).

Page 89 - K. Astala, T. Iwaniec, and E. Saksman, Beltrami operators in the plane, Duke Math. J. 107 (2001), no.