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Hello,

This question is related to Chapter V, lemma 3 on page 54 of Lars Ahlfors' 'Lectures on Quasiconformal mappings' which states :

If $\mu:\mathbb{C}\to \mathbb{D} \in W^{1,p}(\mathbb{C}), p > 2 $ is fixed and sufficiently close to 2 ( because of Calderon-Zygmund theorem used in in the proof of the measurable Riemann Mapping Theorem ), $ L^{\infty} $ norm of $ \mu \le \frac{k-1}{k+1}, k> 0 $ then the solution to the Beltrami equation $f_\bar{z}= \mu . f_z$ is a $\mathcal{C^1}$ diffeomorphism of the complex plane $ \mathbb{C} $.

The above lemma is part of the proof of the measurable Riemann mapping theorem, which generalizes the above lemma with $\mu \in L^\infty(\mathbb{C})$.

I have two questions regarding the above lemma :

1) I went through the proof of the lemma, and it seems like we don't need $\mu$ to be compactly supported.Do we ? [ The reason I am asking is the theorem 1 on page 54 is assumes $\mu$ to be compactly supported and they really use it. Also,you look some other book, e.g. : Imayosgi-Taniguchi's Techmuller Theory , chapter 4, Theorem 4.25, they use the phrase "under the same circumstances of theorem 4.24", which mean they want $\mu$ to be compactly supported, which I don't see why ]

2) Is the following true ? ( I am basically replacing the domain $\mathbb{C}$ by $\mathbb{D}$)

If $\mu:\mathbb{D}\to \mathbb{D} \in W^{1,p}(\mathbb{D}), p > 2 $ and also real-analytic in $\mathbb{D}$, then any solution to the Beltrami equation $f_\bar{z}= \mu . f_z$ is a $\mathcal{C^1}$ diffeomorphism of the open unit disk $ \mathbb{D} $ and also $ f \in C^1({\bar{\mathbb{D} }}) $ [ the last condition is crucial ].

I was trying to answer question no 2 by assuming the following, which might not be correct :

If $\mu \in W^{1,p}(\mathbb{D})$ , then it is Holder continuous on ${\mathbb{D}}$ with Holder exponent $1- \frac{2}{p} $ ( well, I am not assuming this,it follows from the theory of Sobolev Spaces, see Evans' PDE book for example ) and hence is uniformly continuous on $\bar{\mathbb{D} }$.

( probably correct/incorrect ) assumption : Then there exists $ g\in C^\infty(D_2)$ such that $g|_{S^1}= \mu | _{S^1},g \in W^{1,3}(D_2) $ . $D_2$ denotes the ball with radius $2$, centered at $0$.

The reason I was doing this is to transfer the problem to a Beltrami equation on $ \mathbb{C} $, by extending the Beltrami coefficient from $\mathbb{D}$ to $\mathbb{C}$ and the reason I want $L^3$ is that for a finite measure space ( balls of finite radius ) $ L^3 \subset L^p \forall p \le 3 $

Any hints or suggestions or detailed answers for question # 2 ? Thank you !

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  • $\begingroup$ To your first question: It might be true that Ahlfors does not need $\mu$ compactly supported there, but he probably does need $\mu \in W^{1,p}$, so in order to prove it for more general $\mu$ he needs to approximate it anyway, so it might not hurt to throw in compact support at this point. $\endgroup$ Nov 21 '14 at 4:37
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The most comprehensive source for the Beltrami equation in the plane is the book listed below and you should search it for the answers to your questions.

K.Astala, T.Iwaniec, G.Martin, Elliptic partial differential equations and quasiconformal mappings in the plane. Princeton Mathematical Series, 48. Princeton University Press, Princeton, NJ, 2009.

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