$Qn#1 $


: Let $f:U\to V$ be a $K$ quasiconformal homeomorphism ( NOT diffeomorphism ) of plane open subsets of $C$. By my definition of quasiconformality, I mean 1)$f$ is continuous, 2)the weak derivatives $f\_z,f\_\bar{z}$ exist and are $L^2\_{loc}(U)$ functions and 3) $ |\frac{f\_\bar{z}}{f\_z}| \leq k < 1$.


I am trying to prove that $f$ maps sets of measure zero to sets of measure zero. 


Any hints ? Just in case :


I am trying to use Lemma 7.25 [ Chapter : Differentiation ] from Walter Rudin's Real and Complex Analysis which states that :


If $E$ is a Lebesgue measurable subset of $C$ of Lebesgue measure 0,i.e. $m(E)=0$ and if the continuous map $f:E\to C$ has the property that :


$lim sup\_{z\to w} |\frac{f(z)-f(w)}{z-w} |<\infty \forall w \in E$, where $z\to w $ reamining within $E$, then $m(f(E))=0$. Also I am trying to use the upper bound of $L^2$ norms for the difference quotients of $W^{1,2}(U)$ functions in terms of the $L^2$ norms of their gradients . ( L.C. Evans : 5.8.2 ).Acording to conditon 2) gradients are locally $L^2$-bounded. I am not having much success, though !


My guess is I would not need to use condition 3) of quasiconformality, may be I just need to use the weak differentiaility ahthough I am not sure ! 


$Qn#2$:


Also, is there a change-of-variable formula for homeomorphisms in Sobolev spaces ? What is a reference for it ?



How to prove/disprove that quasiconformal maps send measure-zero sets to measure-zero sets
For the first question, see Theorem 3 of Ahlfors' book "[Lectures on quasiconformal mappings](http://books.google.com/books/about/Lectures_on_quasiconformal_mappings.html?id=4oWFH7FPb50C)."


