Abstract
We establish the full range Gagliardo-Nirenberg and the Caffarelli-Kohn-Nirenberg interpolation inequalities associated with Coulomb-Sobolev spaces for the (fractional) derivative 0≤s≤1. As a result, we rediscover known Gagliardo-Nirenberg interpolation type inequalities associated with Coulomb-Sobolev spaces which were previously established in the scale of Hs with 0<s≤1 and extend them for the full range Ws,p with 0≤s≤1 and 1<p<+∞. 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 Lp scale.
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