Sobolev slicing and planar continuity
= Sobolev slicing and planar continuity
{c}
{title2=$C^{0,1-1/q}\text{ on a.e. slice}$}
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 $q=2$.