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This is a question posted in MSE before-https://math.stackexchange.com/questions/3169269/references-on-equivalent-characterization-for-sobolev-spaces-of-functions-of-one:

I cited a result which characterizes Sobolev spaces of functions of one variable as

$ H^p(a,b):= \{ x \in C^{p-1} [a,b]: x^{(p-1)}(t) = \alpha + \int^t_a \Psi ds, \ \alpha \in \mathbb{R}, \Psi \in L^2 \}$,

where $ p \in \mathbb{N} $.

from page 14 of

A. Kirsch: An Introduction to the Mathematical Theory of Inverse Problems. Springer, New York, 1996.

However, the result in this monograph lacks details for proof. Could any researcher help with references with details?

Thanks!

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In your notation $H^k=W^{k,2}$, where $W^{k,p}$ represents the Sobolev space of functions whose derivatives of orders $\leq k$ are in $L^p$.

A function $f\in W^{1,p}(a,b)$ if and only if there is $g\in L^p(a,b)$ and a constant $\alpha$ such that $f(x)=\alpha+\int_a^x g(t)\, dt$ for almost all $x\in (a,b)$. Since $f\in W^{k,p}(a,b)$ if and only if $f'\in W^{k-1,p}(a,b)$ your characterization follows.

For the proof of the above fact see Theorem 1 on p. 163 (in the first edition of the book) in the section Sobolev functions of one variable. You can find this book online.

L. C. Evans, R. F. Gariepy, Measure theory and fine properties of functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992.

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  • $\begingroup$ I can only describe part of my confuse: according to your notations, it seems that the characterization of version C^{p-1} [a,b] lacks the almost everywhere equality. Moreover, in numerical analysis with Sobolev spaces H^{p}(a,b), it is a usual way to give restriction on evaluation at endpoint like x=a or b. It also seems contradicted to the almost everywhere definition you give here? $\endgroup$
    – Yidong Luo
    Commented Mar 31, 2019 at 17:37
  • $\begingroup$ @YidongLuo Sobolev functions are defined almost everywhere by the definition and if you have a fucntion $W^{1,p}(a,b)$, then you can modify it on a set of measure zero so that it is continuous and equal to $f(x)=\alpha+\int_a^x g(t)\, dt$. Just take a book in Sobolev spaces and learn it. This is no place to ask such questions. $\endgroup$
    – Piotr Hajlasz
    Commented Mar 31, 2019 at 17:44
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    $\begingroup$ Get it and thank you for your tolerance. $\endgroup$
    – Yidong Luo
    Commented Mar 31, 2019 at 18:17
  • $\begingroup$ But i still feel some confuse on the claim "Since $ f \in W^{k,p}(a,b) $ if and only if $ f' \in W^{k-1,p}(a,b) $ your characterization follows". Firstly, i have no idea what $ f' $ denote for, almost everywhere or pointwise? Second, i always feel it lacks something to make $ f $ a classical function in $ C^{p-1} [a,b] $ automatically. Sorry for my slow reacting. $\endgroup$
    – Yidong Luo
    Commented Mar 31, 2019 at 18:46
  • $\begingroup$ All the derivatives should be viewed as weak derivatives. If a function is smooth enough, then the derivatives become the standard pointwise derivatives. $\endgroup$
    – Deane Yang
    Commented Apr 1, 2019 at 4:24

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