My question is related to this one Is a Sobolev function absolutely continuous with respect to a.e.segment of line? and its answer, but to my understanding it has not answered the question below.
Let $u \in W^{1,1}(\mathbb{R}^d,\mathbb{R})$ and $\phi \in C_b^\infty(\mathbb{R}^d,\mathbb{R}^d)$. Let $I:= (-\epsilon, \epsilon)$ be such that $\text{Id}+t\phi:\mathbb{R}^d\to \mathbb{R}^d$ is invertible for all $t \in I$.
Is $t\mapsto u((\text{Id}+t\phi)^{-1}(x))$ absolutely continuous for $dx$-a.e. $x\in \mathbb{R}^d$?
Under the assumptions on $\phi$, $t\mapsto (\text{Id}+t\phi)^{-1}(x)$ should be a smooth curve for each $x$. However, for $x_1 \neq x_2$, the curves are not expected to be parallel, so to me it seems a result of type "$u$ is absolutely continuous along almost every curve parallel to any given reasonable curve" would not help.