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 !
