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Let $p\in [1,\infty]$ and take $u\in W^{1,p}(\mathbb{R}^N)$. It is a well know result that $u$ is absolutely continuous (A.C) on a.e. segment of line parallel to the coordinate axes.

It seems to me that there is nothing special about the coordinate axes, so indeed if some direction $v\in \partial B(0,1)$ is fixed, we must have that $u$ is (A.C) on a.e. segment of line with direction $v$. In fact, let $T$ be a orthogonal map which sends $v$ to $e_1=(1,0,...,0)$.

Let $w(x)=u(T^{-1}(x))$. Then, $w\in W^{1,p}(\mathbb{R}^N)$ and $w$ is (A.C) on a.e. segment of line with direction $e_1$. By definition of $w$, we must conclude that $u$ is (A.C) on a.e. segment of line with direction $v$.

My questions are:

1) Is it true that a Sobolev functions is (A.C) for a.e. segment of line? If so, is my argument correct?

2) Can this be generalized for a general family of curves? I mean, assume that $\Gamma$ is a family of (Lipszhitz?) curves. How can I know if $u$ is (A.C) with respect to this family, i.e. $u\circ \gamma$ is (A.C) for $\gamma\in \Gamma$? Is there any kind of measure $\mu$, that we can assign to $\Gamma$, in order to say something like this: $u$ is (A.C) $\mu$ a.e. $\gamma\in \Gamma$?

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  • $\begingroup$ Hi, I am new to these results so I have a few questions. Firstly, why are referring $v \in \partial B(0,1)$ as a direction? Why does it follow that $u$ then must be (A.C.) on a.e. segment of line with direction $v$? Lastly, could you explain the mapping $T$, you say it sends '$T$ to $e_{1}$'?? Thanks, sorry if this trivial. I'm still getting a feel for these topics. $\endgroup$ – user116403 Aug 26 '14 at 13:34
  • $\begingroup$ @JohnJack, fixing $v\in\partial B(0,1)$ is analogous to point your finger in some direction and that's why I called it a direction. For the third question, it is a typo, I will fix it: it sends $v$ to $e_1$. With respect to your second question, note that $u(x)=v(T(x))$, so, if $v$ is (A.C) on a.e. every segment of line with direction $e_1$ then, once $T^{-1}(e_1)=v$, we must conclude that $u$ is (A.C) on a.e. every segment of line with direction $v$. $\endgroup$ – Tomás Aug 26 '14 at 14:52
  • $\begingroup$ @Tomás Typo there, "$u(x) = w(T(x))$, so, if $w$ is (A.C)..." $\endgroup$ – user100431 Aug 26 '14 at 15:54
  • $\begingroup$ @JohnDoe there were plenty of typos and bad notation on my question. I have changed the function $v$ to $w$, because I alread used $v$ for a vector. In my previous comment, we have to change $v$ (the function) for $w$. $\endgroup$ – Tomás Aug 26 '14 at 15:56
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    $\begingroup$ @JohnJack, yes you are right. Because $T$ preserves inner product, it will send every line with direction $v$, to a line with direction $e_1$. $\endgroup$ – Tomás Aug 26 '14 at 20:43
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(1) Yes, your argument is correct. The fact that composition with $T^{-1}$ preserves Sobolev classes also needs to be proved, but the proof is immediate from consideration of what this composition does to Cauchy sequences (wrt $W^{1,p}$ norm) of smooth functions.

(2) Yes, and this generalization is one of fundamental results for the theory of Sobolev functions on metric spaces (which lack the notion of a line segment). The statement is: for every family of curves $\Gamma$ the composition $u\circ \gamma$ is absolutely continuous for all $\gamma\in \Gamma\setminus \Gamma\,'$, where $\Gamma\,'$ is a family of curves with $p$-modulus zero. Obviously, this leads to the question of what is the $p$-modulus of a family of curves... from many available sources, I'll refer you to the survey Sobolev spaces on metric-measure spaces by Hajłasz, specifically Chapter 7 and even more specifically Theorem 7.13.

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  • $\begingroup$ Do you know this book? Is it a good reference on this subject? $\endgroup$ – Tomás Sep 2 '14 at 14:27
  • $\begingroup$ @Tomás Yes, it's well written. Did not you ask something from that book before? I thought you did; it was about the invariance of weak Poincare inequality under biLipschitz maps. $\endgroup$ – user147263 Sep 2 '14 at 14:57
  • $\begingroup$ @Tomás Sorry, this is impossible to answer. This is what PhD advisors are for. I can't direct a PhD thesis over the Internet... $\endgroup$ – user147263 Sep 2 '14 at 17:45
  • $\begingroup$ At least, may I ask you some good references, about the subject of quasiregular mappings and so on? I think it is a really interesting matter and I am willing to study it, however, I have to began from the scratch. There is the book of Seppo Rickman (is it good)? Sometime ago, I learned about some open problems for the $p$-Laplacean Dirichlet eigenvalue problem and I think that, somehow they are related with this area. It seems also that this area is most studied by the Finland school (is this true)? I know you are a busy man and sorry for being so insistent, but I just need a direction. $\endgroup$ – Tomás Sep 6 '14 at 21:17
  • $\begingroup$ @Tomás The book by Rickman is good, but emphasises the method of modulus of curves, which is more geometric and less analytic. The more analytic methods, similar to $p$-Laplace, can be found in: "Space mappings of bounded distortion" by Reshetnyak; "Geometric function theory and non-linear analysis" by Iwaniec&Martin; "Elliptic PDE and quasiconformal mappings in the plane" by Astala, Iwaniec, and Martin. It's true that the area is active in Finland, but it has presence in Spain, Italy, Russia, and the U.S., too. That said, you may want to look into what people around you are studying. $\endgroup$ – user147263 Sep 6 '14 at 21:32

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