Consider $f:R^n\to R$, integrable and weakly differentiable with respect to the $n$-th variable only. We do not know whether weak partial derivatives exist as functions with respect to $x_1,...,x_{n-1}$ (they always exist as distributions though). The weak gradient with respect to $x_n$ is denoted $(\partial/\partial x_n)f$.
- Is it true or false that if $\int..\int |(\partial/\partial x_n) f(x_1,...,x_n)|dx_1,...,dx_n<+\infty$ then for Lebesgue almost every $(x_1,...,x_{n-1})$ in $R^{n-1}$, the function $x_n\to f(x_1,...,x_n)$ is absolutely continuous?
intuition that this may be true.
The ACL property that characterizes Sobolev spaces is that if $f$ is weakly differentiable with respect to all $n$ variables $x_1,...,x_n$ and $\int..\int (|f| + \max_{k=1,...,n}|(\partial/\partial x_k)f |)dx_1,...,dx_n<+\infty$ then the restriction of $f$ to almost every lines parallel to the coordinate axis is absolutely continuous.
The question above asks whether the existence and integrability of $(\partial/\partial x_1)f,...,(\partial/\partial x_{n-1})f$ are necessary for $f$ to be absolutely continuous on almost every lines parallel to the coordinate axis $\{0\}\times...\times \{0\} \times R$.
After some research, the claim is true, see for example the proof of the ACL characterization in section 1.1.3 of Sobolev Spaces with Applications to Elliptic Partial Differential Equations by V. Maz'ya. The same proof is given in http://www-users.math.umn.edu/~poggi008/Talk%201116.pdf.
The proof there focuses on one coordinate, say $x_n$, and only assumes integrability of $(\partial/\partial x_n)f$ so the existence of the other $n-1$ weak derivatives is not required.