Exponential integrability in the spirit of Moser-Trudinger's inequalities of functions with finite non-local, non-convex energy

Abstract

Let $d\ge 1$, $p\ge d$, and let $\Omega$ be a smooth bounded open subset of ${\mathbb{R}}^d$. We prove some exponential integrability in the spirit of Moser-Trudinger's inequalities for measurable functions $u$ defined in $\Omega$ such that $${{\int }_{\Omega }{\int }_{\Omega }}_{|u(x)-u(y)|>\delta }\frac{1}{|x-y{|}^{d+p}}dxdy<+\infty ,$$ for some $\delta >0$. This double integral appeared in characterizations of Sobolev spaces and involved in improvements of the Sobolev inequaliies, Poincaré inequalities, and Hardy inequalities.