Sharp Gagliardo-Nirenberg inequalities in fractional Coulomb-Sobolev spaces

TL;DR

This paper establishes sharp, scaling-invariant Gagliardo-Nirenberg inequalities involving fractional Sobolev norms and Coulomb-type energies, identifying optimal parameter ranges and the impact of radial symmetry on these inequalities.

Contribution

It introduces new sharp inequalities combining fractional Sobolev spaces with Coulomb energies, including optimal parameter ranges and conditions for radial improvements.

Findings

Derived sharp Gagliardo-Nirenberg inequalities with fractional and Coulomb terms.

Identified optimal parameter ranges for the validity of these inequalities.

Showed radial symmetry extends the validity range, with improvements only if >1.

Abstract

We prove scaling invariant Gagliardo-Nirenberg type inequalities of the form $$\|\varphi\|_{L^p(\mathbb{R}^d)}\le C\|\varphi\|_{\dot H^{s}(\mathbb{R}^d)}^{\beta} \left(\iint_{\mathbb{R}^d \times \mathbb{R}^d} \frac{|\varphi (x)|^q\,|\varphi (y)|^q}{|x - y|^{d-\alpha}} dx dy\right)^{\gamma},$$ 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 $\alpha>1$.