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Math people:

This will probably be easy for someone out there. I have functions $u \in C^\infty([0,1])$, $f \in L^1([0,1])$, and $h(t) = \int_0^t f(s)\,ds$. Then $h \in W^{1,1}([0,1])$, right? I'd like to use integration by parts to conclude

$$ \int_0^1 u(t) \frac{d}{dt}(h(t)^2)\,dt = u(t)(h(t))^2|^1_0-\int_0^1 u'(t)(h(t))^2\,dt. $$

Is this valid? I apologize if this is a duplicate. I searched for similar questions and couldn't find one.

Stefan Smith
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    Sobolev functions on the line are exactly those functions which have absolutely continuous representatives. Since $f \in L^1$ it follows that $h$ is $AC$, hence in $W^{1,1}$. The integration by parts formula works because $h^2$ is also $AC$. – Umberto P. Jul 03 '13 at 12:44
  • Thanks, this helps me understand absolutely continuous functions (I never really understood before why people cared about absolutely continuous functions). In proofs of Rademacher's Theorem (any Lipschitz function on $\mathbb{R}^n$ is differentiable a.e.), people usually treat the $n=1$ case as obvious - I guess this is because the conclusion of the theorem applies for AC functions (not just Lipschitz). – Stefan Smith Jul 04 '13 at 17:05
  • Have you found a reference and/or published a paper containing the proof? – Stefan Hante Apr 04 '16 at 08:58

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To answer your first question: Yes, $h \in W^{1,1}(0,1)$ and $h' = f$.

In order to answer questions similar to your second one, density arguments will do a good job:

  • Your assertion holds for smooth $h$ (e.g. $h \in C^\infty([0,1])$).
  • Since $W^{1,1}(0,1) \hookrightarrow C([0,1])$, all terms in your assertion are continuous w.r.t. the $W^{1,1}(0,1)$-norm of $h$.
  • Since $C^\infty([0,1])$ is dense in $W^{1,1}(0,1)$ your assertion follows.
gerw
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  • Thanks. Do you know of a reference where this is done? Someone has undoubtedly done this before. This will be part of a paper. I have several options: (i) don't justify the integration by parts and hope that the reviewer believes it; (ii) justify the integration by parts and give your argument; (iii) justify the integration by parts using a reference. I'll like to use option (iii), since it would save some space. I can obtain Adams's classic book on Sobolev spaces and Evans's excellent book on PDE easily. – Stefan Smith Jul 02 '13 at 14:05
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    Unfortunately, I cannot provide you any reference. However, you could give the above argument abbreviated (i.e., ... which follows by density for all $u \in W^{1,1}(0,1)$. – gerw Jul 02 '13 at 17:30