ON THE MOSER-TRUDINGER INEQUALITY IN FRACTIONAL SOBOLEV-SLOBODECKIJ SPACES

We consider the problem of finding the optimal exponent in the Moser-Trudinger inequality sup{integral(Omega)exp(a vertical bar u vertical bar n/n-s)vertical bar u is an element of(W) over tilde (s,p)(0) (Omega), [u](W)s,p(R-N) <= 1} < +infinity Here Omega is a bounded domain of R-N (N >= 2), s is an element of (0, 1), sp = N, <(W)over tilde>(s,p)(0)(Omega) is a Sobolev-Slobodeckij space, and [.](W)s,p(R-N)=false>(N) is the associated Gagliardo seminorm. 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 alpha is an element of(alpha*(s,N), +infinity).