Sharp Gagliardo–Nirenberg inequalities in fractional Coulomb–Sobolev spaces

2017/10/11 by Jacopo Bellazzini, Marco Ghimenti, Carlo Mercuri +2 · 1 citation
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Geometric Analysis and Curvature Flows #Contact Mechanics and Variational Inequalities

paper · doi:10.1090/tran/7426

Abstract

We prove scaling invariant Gagliardo–Nirenberg type inequalities of the form ‖φ ‖_L^p(\mathbb R^d)≤ C‖φ ‖_ H^s(\mathbb R^d)^β (\iint _\mathbb R^d × \mathbb R^d \frac |φ (x)|^q |φ (y)|^q|x - y|^d-α \textrm dx \textrm dy )^γ , involving fractional Sobolev norms with s>0 and Coulomb type energies with 0<α <d and q≥ 1. We establish optimal ranges of parameters for the validity of such inequalities and discuss the existence of the optimizers. In the special case p=\frac 2dd-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 α >1.

Cited by

• Fractional Schrödinger-Poisson-Slater equations in Coulomb-Sobolev spaces

Related

• Gagliardo-Nirenberg Inequalities in Fractional Coulomb-Sobolev spaces for Radial functions
• Gagliardo-Nirenberg and Caffarelli-Kohn-Nirenberg interpolation inequalities associated with Sobolev-Coulomb spaces
• Gagliardo-Nirenberg type inequalities using fractional Sobolev spaces and Besov spaces
• The sharp Gagliardo--Nirenberg--Sobolev inequality in quantitative form
• Sharp pointwise estimates for functions in the Sobolev spaces Hs(Rn)