We prove scaling invariant Gagliardo–Nirenberg type inequalities of the form (Formula Presented) 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 = 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 α > 1.

Sharp Gagliardo–Nirenberg inequalities in fractional Coulomb–Sobolev spaces / Bellazzini, J.; Ghimenti, M.; Mercuri, C.; Moroz, V.; Van Schaftingen, J.. - In: TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY. - ISSN 0002-9947. - 370:11(2018), pp. 8285-8310. [10.1090/tran/7426]

Sharp Gagliardo–Nirenberg inequalities in fractional Coulomb–Sobolev spaces

Ghimenti M.;Mercuri C.;
2018

Abstract

We prove scaling invariant Gagliardo–Nirenberg type inequalities of the form (Formula Presented) 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 = 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 α > 1.
2018
370
11
8285
8310
Sharp Gagliardo–Nirenberg inequalities in fractional Coulomb–Sobolev spaces / Bellazzini, J.; Ghimenti, M.; Mercuri, C.; Moroz, V.; Van Schaftingen, J.. - In: TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY. - ISSN 0002-9947. - 370:11(2018), pp. 8285-8310. [10.1090/tran/7426]
Bellazzini, J.; Ghimenti, M.; Mercuri, C.; Moroz, V.; Van Schaftingen, J.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11380/1295810
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