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

Let $d\ge 1$, $p\ge d$, and let $\mathrm{\Omega }$ 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 $\mathrm{\Omega }$ such that $${}_{\mathrm{\mid }u(x)-u(y)\mathrm{\mid }>\delta }\frac{1}{\mathrm{\mid }x-y{\mathrm{\mid }}^{d+p}}dxdy<+\mathrm{\infty },$$ for some $\delta >0$. This double integral appeared in characterizations of Sobolev spaces and involved in improvements of the Sobolev inequaliies, Poincar'e inequalities, and Hardy inequalities.