# A regularity question on the Beltrami equation $f_\bar{z} =\mu . f_z$ on $D$

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 !

• 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. Nov 21 '14 at 4:37