Sharp Gagliardo–Nirenberg inequalities in fractional Coulomb–Sobolev spaces
(2018) Transactions of the American mathematical society — Vol. 370, n° 11, p. 8285-8310 (2018)
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- Abstract
- We prove scaling invariant Gagliardo-Nirenberg type inequalities of the form ∥φ∥L^p(R^d)≤C∥φ∥βH^s(R^d)(∬R^d×R^d |φ(x)|^q |φ(y)|^q /|x−y|^(d−α) dx 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 optimisers. In the special case p=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.
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Bellazzini, J., Ghimenti, M., Mercuri, C., Moroz, V., & Van Schaftingen, J. (2018). Sharp Gagliardo–Nirenberg inequalities in fractional Coulomb–Sobolev spaces. Transactions of the American mathematical society, 370(11), 8285-8310. https://doi.org/10.1090/tran/7426 (Original work published 2018)