# American Institute of Mathematical Sciences

August  2018, 38(8): 4019-4040. doi: 10.3934/dcds.2018175

## On fractional Hardy inequalities in convex sets

 1 Dipartimento di Matematica e Informatica, Università degli Studi di Ferrara, Via Machiavelli 35, 44121 Ferrara, Italy 2 Aix-Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, 39 Rue Frédéric Joliot Curie, 13453 Marseille, France 3 Dipartimento di Matematica, Università degli Studi di Bologna, Piazza di Porta San Donato 5, 40126 Bologna, Italy

Received  October 2017 Revised  February 2018 Published  May 2018

We prove a Hardy inequality on convex sets, for fractional Sobolev-Slobodeckiĭ spaces of order $(s, p)$. The proof is based on the fact that in a convex set the distance from the boundary is a superharmonic function, in a suitable sense. The result holds for every $1<p<∞$ and zhongwenzy $0<s<1$, with a constant which is stable as $s$ goes to 1.

Citation: Lorenzo Brasco, Eleonora Cinti. On fractional Hardy inequalities in convex sets. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4019-4040. doi: 10.3934/dcds.2018175
##### References:

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##### References:
The set $\Sigma_\sigma(x)$ and the supporting hyperplane $\Pi_{x'}$
The distance of $y$ from $\partial K$ is smaller than its distance from the hyperplane
The set $K_x$ in the second part of the proof of Theorem 1.1
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