On the Moser-Trudinger inequality in fractional Sobolev-Slobodeckij spaces (Q5915681)

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