How to prove/disprove that quasiconformal maps send measure-zero sets to measure-zero sets - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T12:56:07Z http://mathoverflow.net/feeds/question/66022 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/66022/how-to-prove-disprove-that-quasiconformal-maps-send-measure-zero-sets-to-measure How to prove/disprove that quasiconformal maps send measure-zero sets to measure-zero sets Analysis Now 2011-05-26T06:48:41Z 2011-05-26T15:16:48Z <p>$Qn#1$</p> <p>: 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) $|\frac{f_\bar{z}}{f_z}| \leq k &lt; 1$.</p> <p>I am trying to prove that $f$ maps sets of measure zero to sets of measure zero. </p> <p>Any hints ? Just in case :</p> <p>I am trying to use Lemma 7.25 [ Chapter : Differentiation ] from Walter Rudin's Real and Complex Analysis which states that :</p> <p>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 :</p> <p>$lim sup_{z\to w} |\frac{f(z)-f(w)}{z-w} |&lt;\infty \forall w \in E$, 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 !</p> <p>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 ! </p> <p>$Qn#2$:</p> <p>Also, is there a change-of-variable formula for homeomorphisms in Sobolev spaces ? What is a reference for it ?</p> http://mathoverflow.net/questions/66022/how-to-prove-disprove-that-quasiconformal-maps-send-measure-zero-sets-to-measure/66027#66027 Answer by Tapio Rajala for How to prove/disprove that quasiconformal maps send measure-zero sets to measure-zero sets Tapio Rajala 2011-05-26T07:59:17Z 2011-05-26T08:08:17Z <p>In order to have the <strong>Lusin condition N</strong> (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 $$\frac{|Df|^2}{\log(e+|Df|)} \in L_{\text{loc}}^1(U).$$</p> <p>See [J. Kauhanen, P. Koskela and J. Malý, <em>Mappings of finite distortion: condition N</em>, Michigan Math. J. <strong>49</strong> (2001), no. 1, 169$-$181] <a href="http://projecteuclid.org/euclid.mmj/1008719040" rel="nofollow">http://projecteuclid.org/euclid.mmj/1008719040</a></p> <p>There are change of variables formulas for Sobolev homeomorphisms. See for example [P. Hajĺasz, <em>Change of variables formula under minimal assumptions</em>, Colloq. Math. <strong>64</strong> (1993), 93$-$101] <a href="http://matwbn.icm.edu.pl/ksiazki/cm/cm64/cm64112.pdf" rel="nofollow">http://matwbn.icm.edu.pl/ksiazki/cm/cm64/cm64112.pdf</a> and the references in it.</p> http://mathoverflow.net/questions/66022/how-to-prove-disprove-that-quasiconformal-maps-send-measure-zero-sets-to-measure/66069#66069 Answer by Richard Kent for How to prove/disprove that quasiconformal maps send measure-zero sets to measure-zero sets Richard Kent 2011-05-26T15:16:48Z 2011-05-26T15:16:48Z <p>For the first question, see Theorem 3 of Ahlfors' book "<a href="http://books.google.com/books/about/Lectures_on_quasiconformal_mappings.html?id=4oWFH7FPb50C" rel="nofollow">Lectures on quasiconformal mappings</a>."</p>