**Definition.**
If $U\subset\mathbb{R}$ is open, we say that $u\in {AC}(U)$
if $u$ is absolutely continuous on every compact interval in
$U$. Let $\Omega\subset\mathbb{R}^n$. We say that
$u$ is *absolutely continuous on lines*, $u\in {ACL}(\Omega)$,
if the function $u$ is Borel measurable and for almost every line
$\ell$ parallel to one of the coordinate axes, $u|_\ell\in AC(\Omega\cap\ell)$.
Since absolutely continuous functions in dimension one are differentiable a.e.,
$u\in {ACL}(\Omega)$ has partial derivatives a.e.

> **Theorem.** $f\in L^1_{\rm loc}(\Omega)$ has weak derivative $\nabla f\in L^1_{\rm loc}(\Omega)$ if and only if $f\in ACL(\Omega)$ and the classical partial derivatives (which exist a.e.) are in $L^1_{\rm loc}(\Omega)$. Moreover the classical partial derivatives of $f$ which exists a.e., equal to the weak partial derivatives.

**Remark.** Functions that are equal a.e. are identified so by writing $f\in ACL(\Omega)$ we mean that $f$ equals a.e. to a function that belongs to $ACL(\Omega)$.

The above result is Theorem 4.21 in [EG] or Theorem 2.23 in [H] or Theorem 1 p. 4 and Theorem 2 p. 6 in [M].

**[EG] L. C. Evans, R. F. Gariepy,** *Measure theory and fine properties of functions.* Revised edition.
Textbooks in Mathematics. CRC Press, Boca Raton, FL, 2015.

**[H] P. Hajłasz** [Non-linear elliptic partial differential equations.][1], 2010.

**[M] V. Maz'ya,** *Sobolev spaces with applications to elliptic partial differential equations.* Second, revised and augmented edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 342. Springer, Heidelberg, 2011. 


  [1]: https://www.pitt.edu/~hajlasz/Notatki/Cortona%20Lectures.pdf