{"paper":{"title":"Sharp Gagliardo-Nirenberg inequalities in fractional Coulomb-Sobolev spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.AP","math.MP"],"primary_cat":"math.FA","authors_text":"Carlo Mercuri, Jacopo Bellazzini, Jean Van Schaftingen, Marco Ghimenti, Vitaly Moroz","submitted_at":"2016-12-01T13:46:45Z","abstract_excerpt":"We prove scaling invariant Gagliardo-Nirenberg type inequalities of the form $$\\|\\varphi\\|_{L^p(\\mathbb{R}^d)}\\le C\\|\\varphi\\|_{\\dot H^{s}(\\mathbb{R}^d)}^{\\beta} \\left(\\iint_{\\mathbb{R}^d \\times \\mathbb{R}^d} \\frac{|\\varphi (x)|^q\\,|\\varphi (y)|^q}{|x - y|^{d-\\alpha}} dx dy\\right)^{\\gamma},$$ 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 optimisers. In the special case $p=\\frac{2d}{d-2s}$ our results include a new refinement "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.00243","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}