Sharp Gagliardo–Nirenberg inequalities in fractional Coulomb–Sobolev spaces

We prove scaling invariant Gagliardo-Nirenberg type inequalities of the form ∥φ∥L^p(R^d)≤C∥φ∥βH^s(R^d)(∬R^d×R^d |φ(x)|^q |φ(y)|^q /|x−y|^(d−α) dx dy)^γ, involving fractional Sobolev norms with s>0 and Coulomb type energies with 01