Sharp Gagliardo-Nirenberg inequalities in fractional Coulomb-Sobolev spaces

We prove scaling invariant Gagliardo-Nirenberg type inequalities of the form $$|arphi|_L^p(mathbbR^d)le C|arphi|_dot H^s(mathbbR^d)^eta left(iint_mathbbR^d imes mathbbR^d rac|arphi (x)|^q,|arphi (y)|^q|x - y|^d-alpha dx dy ight)^gamma,$$ involving fractional Sobolev norms with $s>0$ and Coulomb type energies with $01$.