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 $$ \mathop{\int_{\Omega} \int_{\Omega}}_{|u(x) - u(y)| > \delta} \frac{1}{|x-y|^{d+p}} \, dx \, dy < + \infty, $$ 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.