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

2021/10/25 by Mallick, Arka, Nguyen, Hoai-Minh
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2110.13198

Abstract

We establish the full range Gagliardo-Nirenberg and the Caffarelli-Kohn-Nirenberg interpolation inequalities associated with Sobolev-Coulomb spaces for the (fractional) derivative 0 ≤ s ≤ 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 ≤ 1 and extend them for the full range W^{s, 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 L^p scale.

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