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

We give a contribution to the problem of finding the optimal exponent in the Moser-Trudinger inequality in the fractional Sobolev-Slobodeckij space (W) over tildeo(s,p)(Omega), where Omega subset of R-N is a bounded domain, is an element of (0, 1), and sp = N. We exhibit an explicit exponent alpha*(s, N) > 0, which does not depend on Omega, such that the Moser-Trudinger inequality does not hold true for is an element of (alpha*(s, N), +infinity).