Sobolev fundamental theorem of calculus on lines

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 satisfieswhenever the line segment lies in $U$.

For $u\in W^{1,q}((0,1{)}^2)$ with $q>1$, almost every coordinate-line slice has an absolutely continuous representative and is Hölder continuous with exponent $1-1/q$. Slice bounds need not be uniform. Global Hölder continuity follows from Morrey's inequality only when $q>2$, with exponent $1-2/q$; unbounded examples exist at and below $q=2$.