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We prove scaling invariant Gagliardo-Nirenberg type inequalities of the form$$\|\varphi\|_{L^p(\mathbb{R}^d)}\le C\|\varphi\|_{\dotH^{s}(\mathbb{R}^d)}^{\beta} \left(\iint_{\mathbb{R}^d \times \mathbb{R}^d}\frac{|\varphi (x)|^q\,|\varphi (y)|^q}{|x - y|^{d-\alpha}} dxdy\right)^{\gamma},$$ involving fractional Sobolev norms with $s>0$ and Coulombtype energies with $0<\alpha<d$ and $q\ge 1$. We establish optimal ranges ofparameters for the validity of such inequalities and discuss the existence ofthe optimisers. In the special case $p=\frac{2d}{d-2s}$ our results include anew refinement of the fractional Sobolev inequality by a Coulomb term. We alsoprove that if the radial symmetry is taken into account, then the ranges ofvalidity of the inequalities could be extended and such a radial improvement ispossible if and only if $\alpha>1$.

Sharp Gagliardo-Nirenberg inequalities in fractional Coulomb-Sobolev spaces