Hi, 


On page 120, chapter 4, proposition 4.2.7 in Hubbard's Teichmuller Theory book, volume 1, he proves :


Let $U,V$ be open in $C, f:U \to V $ be a homeomorphism and the restriction of $f$ on $U \backslash l$ is $K$ quasiconformal, where $l$ is a line in $U$. Then $f$ is $K$-q.c. in $U$.


FYI: his definition 4.1.1 ( page 112 ) of a $K$-q.c. map $f:U\to V$ is : 


1) $f$ is a homeomorphism.


2) Distributional/weak derivatives $f\_z,f\_\bar{z}$ of $f$ exist almost everywhere and these derivatives are in $L^2\_{loc}(U)$.


3)$|\frac{f\_\bar{z}}{f\_z}|\leq k = \frac {K-1}{K+1} $ for some $K\geq1$.


Now for the proof of Proposition 4.2.7, isn't it $ENOUGH$ to prove/check that the derivatives $f\_z, f\_\bar{z}$ are in $L^2(K) \forall K$ compact subset of $U$,i.e. condition (2), which does not readily follow from that they are in $L^2\_{loc}(K') \forall K' $ compact subsets oi $U \backslash l$. This he proves by using condition (3) stated above .


But what else is there to prove, since we only care about existence of partial derivatives on a full measure set, we do NOT need to prove anything else apart from condition (2), right ? So, what does he do after proving condition (2), in page 120-121 ?


Thanks and Happy Memorial Day!



A quick and elementary question from Hubbard's Teichmuller Theory : Volume I
All that Hubbard has shown up to that point is that the functions $f\_z$ and $f\_{\bar{z}}$ are locally in $L^2(R)$, i.e. they are defined almost everywhere and are square integrable in $R$ as functions. But all we know at this point is that they are the distributional derivatives of $f$ away from $l$. What he shows after that is that they are actually the distributional derivatives of $f$ in all of $R$ rather than just $R-l$. I can understand the confusion, and I am surprised that Hubbard does not make this more explicit (he is generally very careful about these kinds of issues, both in writing and in conversations).


Near the top of the page, Hubbard remarks that $[\widetilde{Df}]\in L^1(R)$ rather than just $R-l$. This is a key point. He writes that this allows him to justify applying Fubini's theorem, but he also needs this fact to apply the Lebesgue dominated convergence theorem. I'll point out exactly where.


On the series of equalities in 4.2.14, the second equality follows from Lebesgue dominated convergence since $f$ and $\phi$ are continuous and hence bounded on $R$; similarly for the third equality. The fourth equality follows from the fact that away from $l$ our function $f$ has a weak derivative locally in $L^2$ and hence $L^1$. There are now two limits to compute in this sum. The second one is zero since $f$ is continuous and the Lebesgue dominated convergence theorem applies. One may now write the remaining double integral as a single integral over $R\_\epsilon$ by Fubini and then push the limit into the integral by Lebesgue dominated convergence (these two steps are where Hubbard's remark is used).


These equalities guarantee that $f\_x$ and $f\_y$ are indeed the distributional derivatives in $R$. And now, exactly as you've said, this, along with the computations he has already made, give you exactly what you need. I hope this explains everything for you.


