A note on zero sets of fractional sobolev functions with negative power of integrability

TL;DR

This paper extends Poincaré-type inequalities to fractional Sobolev spaces, providing Hausdorff dimension estimates for zero sets of functions with integrable inverses, and offers an elementary proof for a suboptimal estimate.

Contribution

It generalizes existing inequalities to fractional Sobolev spaces and derives new Hausdorff dimension bounds for zero sets of such functions.

Findings

Extended Poincaré inequality to fractional Sobolev spaces.

Derived Hausdorff dimension estimates for zero sets.

Provided an elementary proof for a suboptimal estimate.

Abstract

We extend a Poincar\'{e}-type inequality for functions with large zero-sets by Jiang and Lin to fractional Sobolev spaces. As a consequence, we obtain a Hausdorff dimension estimate on the size of zero sets for fractional Sobolev functions whose inverse is integrable. Also, for a suboptimal Hausdorff dimension estimate, we give a completely elementary proof based on a pointwise Poincar\'{e}-style inequality.