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$Qn#1$

: 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 :

$lim sup_{z\to w} \frac{f(z)-f(w)}{z-w} |\frac{f(z)-f(w)}{z-w} |<\infty \forall w \in E$, 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 ?

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$Qn#1$

: 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 )Without .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 ?

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