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

2024/07/17 by Csató, Gyula, Roy, Prosenjit
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2407.12439

Abstract

In this paper we prove a fractional version of a Caffarelli-Kohn-Nirenberg type interpolation inequality on hypersurfaces M⊂\Rⁿ⁺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 Cabré, Cozzi, and the first author, and a new fractional Hardy inequality on M. Our method, when restricted to the plane case M=\Rⁿ, 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.

Related

• A fractional Michael-Simon Sobolev inequality on convex hypersurfaces
• Fractional Sobolev and Hardy-Littlewood-Sobolev inequalities
• A geometrical approach to the sharp Hardy inequality in Sobolev-Slobodecki\uı spaces
• Universal Hardy-Sobolev inequalities on hypersurfaces of Euclidean space
• On fractional Hardy inequalities in convex sets