{"paper":{"title":"Extended Caffarelli-Kohn-Nirenberg inequalities, and remainders, stability, and superweights for $L^{p}$-weighted Hardy inequalities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Durvudkhan Suragan, Michael Ruzhansky, Nurgissa Yessirkegenov","submitted_at":"2017-01-05T11:25:17Z","abstract_excerpt":"In this paper we give an extension of the classical Caffarelli-Kohn-Nirenberg inequalities: we show that for $1<p,q<\\infty$, $0<r<\\infty$ with $p+q\\geq r$, $\\delta\\in[0,1]\\cap\\left[\\frac{r-q}{r},\\frac{p}{r}\\right]$ with $\\frac{\\delta r}{p}+\\frac{(1-\\delta)r}{q}=1$ and $a$, $b$, $c\\in\\mathbb{R}$ with $c=\\delta(a-1)+b(1-\\delta)$, and for all functions $f\\in C_{0}^{\\infty}(\\mathbb{R}^{n}\\backslash\\{0\\})$ we have\n  $$ \\||x|^{c}f\\|_{L^{r}(\\mathbb{R}^{n})} \\leq \\left|\\frac{p}{n-p(1-a)}\\right|^{\\delta} \\left\\||x|^{a}\\nabla f\\right\\|^{\\delta}_{L^{p}(\\mathbb{R}^{n})} \\left\\||x|^{b}f\\right\\|^{1-\\delta}_"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.01280","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"}