On the Moser–Trudinger inequality in fractional Sobolev–Slobodeckij spaces

Abstract

We give a contribution to the problem of finding the optimal exponent in the Moser–Trudinger inequality in the fractional Sobolev–Slobodeckij space ${\tilde{W}}_0^{s,p}(\Omega )$, where $\Omega \subset {\mathbb{R}}^N$ is a bounded domain, $s\in (0,1)$, and $sp=N$. We exhibit an explicit exponent ${\alpha }_{s,N}^{\ast }>0$, which does not depend on $\Omega$, such that the Moser–Trudinger inequality does not hold true for $\alpha \in ({\alpha }_{s,N}^{\ast },+\infty )$.