{"paper":{"title":"On some nonlinear extensions of the Gagliardo-Nirenberg inequality with applications to nonlinear eigenvalue problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Agnieszka Ka{\\l}amajska, Jan Peszek","submitted_at":"2011-04-11T15:02:22Z","abstract_excerpt":"We derive inequality [\\int_{\\r} |f^{'}(x)|^ph(f(x))dx \\le (\\sqrt{p-1})^p\\int_{\\r}(\\sqrt{|f^{\"}(x){\\cal T}_h(f(x))|})^ph(f(x))dx,] where $f$ belongs locally to Sobolev space $W^{2,1}$ and $f^{'}$ has bounded support. Here $h(...)$ is a given function and ${\\cal T}_h(...)$ is its given transform, it is independent of $p$. In case when $h\\equiv 1$ we retrieve the well known inequality: (\\int_{\\r} |f^{'}(x)|^pdx \\le (\\sqrt{p-1})^p \\int_{\\r}(\\sqrt{|f^{\"}(x)f(x)|})^pdx.) Our inequalities have form similar to the classical second order Oppial inequalites. They also extend certain class of inequalitie"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1104.1967","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"}