  We establish the full range Gagliardo-Nirenberg and the
Caffarelli-Kohn-Nirenberg interpolation inequalities associated with
Sobolev-Coulomb spaces for the (fractional) derivative $0 \leq s \leq 1$. As a
result, we rediscover known Gaglairdo-Nirenberg interpolation type inequalities
associated with Sobolev-Coulomb spaces which were previously established in the
scale of $H^{s}$ with $0 < s \leq 1$ and extend them for the full range $W^{s,
p}$ with $0\leq s \leq 1$ and $1 < p < + \infty$. Using these newly established
weighted inequalities, we derive a new family of one body Hardy-Lieb-Thirring
inequalities and use it to establish a new family of many body
Hardy-Lieb-Thirring inequalities with a strong repulsive interaction term in
$L^p$ scale.
