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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","authors_text":"Aleksey Kostenko","cross_cats":["math.AP","math.CA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2012-07-11T10:38:32Z","title":"The similarity problem for indefinite Sturm-Liouville operators and the HELP inequality"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.2586","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:92ec6d794ea084fcde7626d2389629e38c28b4db7275d03183506cdada002fc4","target":"record","created_at":"2026-05-18T03:15:06Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"f778fb066eb2bfeed5cf062eee41d30eedf6825d89b6dddae990a64356316473","cross_cats_sorted":["math.AP","math.CA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2012-07-11T10:38:32Z","title_canon_sha256":"00a907b7c8f7ba6297d15285932fba1645fcef103f8f69f4c0b2cdc0bd61ff38"},"schema_version":"1.0","source":{"id":"1207.2586","kind":"arxiv","version":1}},"canonical_sha256":"68a537b61a45d921683ba86a4eaca5c5ba4b6ec64912466d9caa74a5d7e7305f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"68a537b61a45d921683ba86a4eaca5c5ba4b6ec64912466d9caa74a5d7e7305f","first_computed_at":"2026-05-18T03:15:06.536484Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:15:06.536484Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"DgG/AKzacgN39onOjioWsZqXQYlRJu6V5wos3dBQ7VM455tq275KYoGEX2pbIpCc6x+jGBZq0XzUJPGDeF6UBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:15:06.537208Z","signed_message":"canonical_sha256_bytes"},"source_id":"1207.2586","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:92ec6d794ea084fcde7626d2389629e38c28b4db7275d03183506cdada002fc4","sha256:c81cb3290d873cc675dfd316815b648b5037b7ece074db01d40d8aa9f236ea09"],"state_sha256":"c43a8346ef871075b89c66060cd2c586c37c85833dc325c96b6d01e96a05fdeb"}