A fractional Hardy-Sobolev inequality of Michael-Simon type on convex hypersurfaces

TL;DR

This paper establishes a new fractional Hardy-Sobolev inequality on convex hypersurfaces, connecting recent Sobolev inequalities with Hardy inequalities via a universal constant and fractional mean curvature.

Contribution

It introduces a fractional interpolation inequality on convex hypersurfaces involving fractional mean curvature, unifying and extending previous Sobolev and Hardy inequalities.

Findings

Proves a fractional Hardy-Sobolev inequality with a universal constant.

Provides a new proof of the fractional Hardy inequality in the plane case.

Establishes a perimeter inequality for convex hypersurfaces.

Abstract

In this paper we prove a fractional version of a Caffarelli-Kohn-Nirenberg type interpolation inequality on hypersurfaces $M\subset\R^{n+1}$ which are boundaries of convex sets. The inequality carries a universal constant independent of $M$ and involves the fractional mean curvature of $M.$ In particular, it interpolates between the fractional Micheal-Simon Sobolev inequality recently obtained by Cabr\'e, Cozzi, and the first author, and a new fractional Hardy inequality on $M$. Our method, when restricted to the plane case $M=\R^n$, gives a new simple proof of the fractional Hardy inequality. To obtain the fractional Hardy inequality on a hypersurface, we establish an inequality which bounds a weighted perimeter of $M$ by the standard perimeter of $M$ (modulo a universal constant), and which is valid for all convex hypersurfaces $M$.