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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Sharp Gagliardo-Nirenberg inequalities in fractional Coulomb-Sobolev spaces


Authors: Jacopo Bellazzini, Marco Ghimenti, Carlo Mercuri, Vitaly Moroz and Jean Van Schaftingen
Journal: Trans. Amer. Math. Soc. 370 (2018), 8285-8310
MSC (2010): Primary 46E35; Secondary 39B62, 35Q55
DOI: https://doi.org/10.1090/tran/7426
Published electronically: August 9, 2018
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Abstract: We prove scaling invariant Gagliardo-Nirenberg type inequalities of the form

$\displaystyle \Vert\varphi \Vert _{L^p(\mathbb{R}^d)}\le C\Vert\varphi \Vert _{... ...\vert^q}{\vert x - y\vert^{d-\alpha }} {\rm d}x \; {\rm d}y \right )^{\gamma },$    

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 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$.

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Additional Information

Jacopo Bellazzini
Affiliation: Università di Sassari, Via Piandanna 4, 07100 Sassari, Italy
Email: jbellazzini@uniss.it

Marco Ghimenti
Affiliation: Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo 5, 56100 Pisa, Italy
Email: marco.ghimenti@dma.unipi.it

Carlo Mercuri
Affiliation: Department of Mathematics, Swansea University, Singleton Park, Swansea, SA2 8PP, Wales, United Kingdom
Email: C.Mercuri@swansea.ac.uk

Vitaly Moroz
Affiliation: Department of Mathematics, Swansea University, Singleton Park, Swansea, SA2 8PP, Wales, United Kingdom
Email: V.Moroz@swansea.ac.uk

Jean Van Schaftingen
Affiliation: Université Catholique de Louvain, Institut de Recherche en Mathématique et Physique, Chemin du Cyclotron 2 bte L7.01.01, 1348 Louvain-la-Neuve, Belgium
Email: jean.vanschaftingen@uclouvain.be

DOI: https://doi.org/10.1090/tran/7426
Keywords: Interpolation inequalities, fractional Sobolev inequality, Coulomb energy, Riesz potential, radial symmetry
Received by editor(s): June 2, 2017
Received by editor(s) in revised form: September 25, 2017
Published electronically: August 9, 2018
Additional Notes: The first author and second author were supported by GNAMPA 2016 project “Equazioni non lineari dispersive”.
The second author was partially supported by P.R.A. 2016, University of Pisa.
The fifth author was supported by the Projet de Recherche (Fonds de la Recherche Scientifique–FNRS) T.1110.14 “Existence and asymptotic behavior of solutions to systems of semilinear elliptic partial differential equations”.
Article copyright: © Copyright 2018 American Mathematical Society

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