Sharp Gagliardo–Nirenberg inequalities in fractional Coulomb–Sobolev spaces

We prove scaling invariant Gagliardo–Nirenberg type inequalities of the form (Formula Presented) involving fractional Sobolev norms with s > 0 and Coulomb type energies with 0 < α < 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 α > 1.