Question

$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 ?

Answer 1

For the first question, see Theorem 3 of Ahlfors' book "Lectures on quasiconformal mappings."

Answer 2

In order to have the Lusin condition N (the property that sets of Lebesgue measure zero get mapped to sets of Lebesgue measure zero) for a Sobolev homeomorphism in the plane the assumption you need on the integrability of the differential is slightly less than power 2. Namely, it is sufficient to assume $$ \frac{|Df|^2}{\log(e+|Df|)} \in L_{\text{loc}}^1(U). $$

See [J. Kauhanen, P. Koskela and J. Malý, Mappings of finite distortion: condition N, Michigan Math. J. 49 (2001), no. 1, 169$-$181] http://projecteuclid.org/euclid.mmj/1008719040

There are change of variables formulas for Sobolev homeomorphisms. See for example [P. Hajĺasz, Change of variables formula under minimal assumptions, Colloq. Math. 64 (1993), 93$-$101] http://matwbn.icm.edu.pl/ksiazki/cm/cm64/cm64112.pdf and the references in it.