TITLE: quasiconformal across the real line
QUESTION [0 upvotes]: My question is from page 194,line 12 from below, in the book Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane, written by Astala, Iwaniec and Martin.
Let $F:\mathbb{C}\to \mathbb{C}$ be a homeomorphism which is quasiconformal off $\mathbb{R}$. The authors want to show that $F$ is quasiconformal in the plane.
Quote from the argument

Absolutely continuity of $F$ on almost all horizontal lines is clear...
For vertical lines, the only issue is near $\mathbb{R}$, where we may use the uniform continuity of $F$ on any compact neighborhood of a closed subinterval of $\mathbb{R}$.

I don't understand the last sentence. Can anyone explain that to me? I think the authors use the following definition of quasiconformal map.
A homeomorphism $f:\Omega\to \Omega'$ is quasiconformal if

$f$ is absolutely continuous on almost everyline in $\Omega$.
$|\max_\alpha \partial_\alpha f(z)|\leq K |\min_\alpha \partial_\alpha f(z)|$ for a.e. $z\in \Omega$

If we use conformal removability of $\mathbb{R}$ then it is ok. But I don't think that's what the authors meant. Thanks.
Add: More specifically, if $F$ is absolutely continuous on every compact sets of $\{iy: y>0\}$ and $\{iy: y<0\}$ and $F$ is uniformly continuous in a neighborhood of $0$, then why is $F$ absolutely continuous on $i\mathbb{R}$?

REPLY [3 votes]: I think the argument they used is very clear. As you have noticed already, one can easily verify the analytic definition. For property 2 in the analytic definition, it is clear. So one only needs to verify property 1. 
We know that $f\in W^{1,1}_{loc}$ if and only if $f$ is absolutely continuous on almost every line segment that is parallel to the coordinate axis. To verify the ACL property, one only need to show the following:
If $E$ is a line segment intersecting the real line $\mathbb{R}$ such that $f$ is ACL on $E\cap \mathbb{H}^+$ and ACL on $E\cap \mathbb{H}^-$, then $f$ is ACL on $E$.
This is clear by definition. Since ACL means that for any fixed scale, there exists a partition of $E$ such that the variation of $f$ is smaller than the given scale. Around the singularity point on the real line, the uniform continuity of $f$ on each compact neighborhood implies that one can make the variation small by taking the neighborhood small. Then away from the neighborhood, $f$ is ACL and hence the total variation is small as well.
A philosophical way ot thinking about this from the point view of ACL mapping is the following: First of all, one notices that for a continuous function of bounded variation defined on the real line $\mathbb{R}$, $f$ is ACL if and only if $f$ maps sets of (Lebesgue) measure zero to sets of (Lebesgue) measure zero (the so-called Lusin's Condition N). Second, for a homeomorphism $f$ of bounded variation between subsets of the real line $\mathbb{R}$, the essential way for $f$ fails to satisfy Lusin Condition N is that there is a cantor set of measure zero that gets mapped to a set of positive measure. Moreover, from the point view of the construction of counter-examples, the singular cantor set must be non-trivial, it is of Lebesgue measure zero but has positive Hausdorff dimension. In our case, $f$ is homeomorphism and BV, the only singularity is one point on the real line $\mathbb{R}$ and so it must be ACL by the explaination above. This applies to the removability result of more general homeomorphism of finite distortion. You may read the following thesis for an extension of the above (analytic) philosophy:
Rajala, Kai Mappings of finite distortion: removable singularities. Dissertation, University of Jyväskylä, Jyväskylä, 2003. Report. University of Jyväskylä Department of Mathematics and Statistics, 91. University of Jyväskylä, Jyväskylä, 2003. 
As you already observed, for the removability problem in the category of quasiconformal mappings, you may use the geometric definition of quasiconformality as well. This is clear simply notice that the real line $\mathbb{R}$ is of zero conformal capacity and sets of zero capacity are removable for quaisconformal/quasiregular mappings. See also the thesis of Kai for the removablity result from the geometric point of view. 
I wish this at least gives you a brief view of this kind of problems in geometric function theory.
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