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

Abstract In this paper we prove a fractional version of a Caffarelli–Kohn–Nirenberg type interpolation inequality on hypersurfaces $$M\subset \mathbb {R}^{n+1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>M</mml:mi> <mml:mo>⊂</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> 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 Cabré, Cozzi, and the first author, and a new fractional Hardy inequality on M . Our method, when restricted to the plane case $$M=\mathbb {R}^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>M</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> </mml:math> , 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 .