Preprint

Inserted: 13 oct 2026
Last Updated: 14 oct 2026

Pages: 6
Year: 2026
Notes:

We show that the fractional Michael--Simon inequality does not hold for general smooth hypersurfaces and for the range of $s \in (0, \frac{n-1}{n})$ in the case $p =1$. We construct a family of smooth hypersurfaces, whose fractional mean curvature is nonnegative everywhere, which violates the inequality in the limit. The same construction yields a counterexample for every $1 \leq p < \infty$.

Keywords: Michael-Simon inequality, fractional perimeter, Allard inequality, fractional curvature