{"paper":{"title":"On the best constant for Gagliardo-Nirenberg interpolation inequalities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jian-Guo Liu, Jinhuan Wang","submitted_at":"2017-12-29T12:58:04Z","abstract_excerpt":"In this paper we derive the best constant for the following Gagliardo-Nirenberg interpolation inequality\n  \\begin{eqnarray*} \\|u\\|_{L^{m+1}}\\leq C_{q,m,p} \\|u\\|^{1-\\theta}_{L^{q+1}}\\|\\nabla u\\|^{\\theta}_{L^p},\\quad \\theta=\\frac{pd(m-q)}{(m+1)[d(p-q-1)+p(q+1)]}, \\end{eqnarray*} where parameters $q,m,p$ respectively belong to the following two ranges:\n  (i) $p>d\\geq 1$, $q\\geq0$ and $m=\\infty$. That shows $L^{\\infty}$-type Gagliardo-Nirenberg interpolation inequality.\n  (ii) $p>\\max\\{1,\\frac{2d}{d+2}\\}$, $0\\leq q<\\sigma-1$, and $q<m<\\sigma$, where $\\sigma$ is defined by $ \\sigma:= \\frac{(p-1)d+p"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.10208","kind":"arxiv","version":1},"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"}