Sharp Gagliardo-Nirenberg inequalities in fractional Coulomb-Sobolev spaces

Abstract

We prove scaling invariant Gagliardo-Nirenberg type inequalities of the form $$\Vert \phi {\Vert }_{L^p({\mathbb{R}}^d)}\le C\Vert \phi {\Vert }_{{\dot{H}}^s({\mathbb{R}}^d)}^{\beta }{\left({\iint }_{{\mathbb{R}}^d\times {\mathbb{R}}^d}\frac{|\phi (x){|}^q|\phi (y){|}^q}{|x-y{|}^{d-\alpha }}dxdy\right)}^{\gamma },$$ involving fractional Sobolev norms with $s>0$ and Coulomb type energies with $0<\alpha <d$ and $q\ge 1$. We establish optimal ranges of parameters for the validity of such inequalities and discuss the existence of the optimisers. In the special case $p=\frac{2d}{d-2s}$ our results include a new refinement of the fractional Sobolev inequality by a Coulomb term. We also prove that if the radial symmetry is taken into account, then the ranges of validity of the inequalities could be extended and such a radial improvement is possible if and only if $\alpha >1$.

27 pages, introduction and references updated

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