The <Sobolev fundamental theorem of calculus on lines> gives, for almost every $x$,
$$
\Delta_i^hu(x)=\int_0^1D_i u(x+the_i)\,dt.
$$
Apply the <Minkowski integral inequality> and translation invariance of <Lebesgue measure>:
$$
\lVert\Delta_i^hu\rVert_{L^2(V)}
\leq\int_0^1\lVert D_i u(\,cdot+the_i)\rVert_{L^2(V)}dt
\leq\lVert D_i u\rVert_{L^2(U)}.
$$
The restriction on $h$ ensures that every translated copy used above lies in $U$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 105 / 1 / c / i / Solution

{c}

If $u\in W^{1,p}(U)$, then after modification on a null set its restriction to almost every line parallel to a coordinate axis is absolutely continuous and satisfies
$$
u(x+he_i)-u(x)=\int_0^hD_i u(x+se_i)\,ds
$$
whenever the line segment lies in $U$.


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 Incoming links: Sobolev fundamental theorem of calculus on lines