SHARP GAGLIARDO-NIRENBERG INEQUALITIES IN FRACTIONAL COULOMB-SOBOLEV SPACES

We prove scaling invariant Gagliardo-Nirenberg type inequalities of the form parallel to phi parallel to(Lp(Rd)) <= C parallel to phi parallel to(beta)(<(H)over dot>s (Rd)) (integral integral(Rd x Rd) vertical bar phi(x)vertical bar(q)vertical bar phi(y)vertical bar(q)/vertical bar x - y vertical bar(d - alpha) dx dy)(gamma), involving fractional Sobolev norms with s > 0 and Coulomb type energies with 0 < alpha < d and q >= 1. We establish optimal ranges of parameters for the validity of such inequalities and discuss the existence of the optimizers. In the special case p = 2d/d-2s our results include a new refinement of the fractional Sobolev inequality by a Coulomb term. We also prove that if the radial symmetry is taken into account, then the ranges of validity of the inequalities could be extended and such a radial improvement is possible if and only if alpha > 1.