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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 $f_z,f_\bar{Z}$ 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 of 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!
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A quick and elementary question from Hubbard's Teichmuller Theory : Volume I

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 of $U \backslash l$. 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 !