Sharp Gagliardo–Nirenberg inequalities in fractional Coulomb–Sobolev spaces

Abstract:

We prove scaling invariant Gagliardo–Nirenberg type inequalities of the form \begin{equation*} \|\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 }} \textrm {d}x \; \textrm {d}y \right )^{\gamma }, \end{equation*} 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 optimizers. 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 $\alpha >1$.