Sharp Gagliardo-Nirenberg inequalities in fractional Coulomb-Sobolev spaces

We prove scaling invariant Gagliardo-Nirenberg type inequalities of the form $$\mathrm{\parallel }\phi {\mathrm{\parallel }}_{L^p({\mathbb{R}}^d)}\le C\mathrm{\parallel }\phi {\mathrm{\parallel }}_{{\dot{H}}^s({\mathbb{R}}^d)}^{\beta }{\left({\iint }_{{\mathbb{R}}^d\times {\mathbb{R}}^d}\frac{\mathrm{\mid }\phi (x){\mathrm{\mid }}^q\mathrm{\mid }\phi (y){\mathrm{\mid }}^q}{\mathrm{\mid }x-y{\mathrm{\mid }}^{d-\alpha }}dxdy\right)}^{\gamma },$$ involving fractional Sobolev norms with $s>0$ and Coulomb type energies with $0<\alpha <d$ and $q\ge 1$. We establish optimal ranges of parameters for the validity of such inequalities and discuss the existence of the optimisers. In the special case $p=\frac{2d}{d-2s}$ 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 $\alpha >1$.