{"paper":{"title":"The similarity problem for indefinite Sturm-Liouville operators and the HELP inequality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.CA"],"primary_cat":"math.SP","authors_text":"Aleksey Kostenko","submitted_at":"2012-07-11T10:38:32Z","abstract_excerpt":"We study two problems. The first one is the similarity problem for the indefinite Sturm-Liouville operator \\[ A=-(\\sgn\\, x)\\frac{d}{wdx}\\frac{d}{rdx} \\] acting in $L^2_{w}(-b,b)$. It is assumed that $w,r\\in L^1_{\\loc}(-b,b)$ are even and positive a.e. on $(-b,b)$.\n  The second object is the so-called HELP inequality \\[(\\int_{0}^b\\frac{1}{\\tilde{r}}|f'|\\, dx)^2 \\le K^2 \\int_{0}^b|f|^2\\tilde{w}\\,dx\\int_{0}^b\\Big|\\frac{1}{\\tilde{w}}\\big(\\frac{1}{\\tilde{r}}f'\\big)'\\Big|^2\\tilde{w}\\, dx, \\] where the coefficients $\\tilde{w},\\tilde{r}\\in L^1_{\\loc}[0,b)$ are positive a.e. on $(0,b)$.\n  Both problems"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.2586","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"}