Gagliardo-Nirenberg and Caffarelli-Kohn-Nirenberg interpolation inequalities associated with Coulomb-Sobolev spaces

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