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Suppose $p:\\mathbb{R}\\to(1,\\infty)$ is a slowly oscillating exponent such that the Cauchy singular integral operator $S$ is bounded on the variable Lebesgue space $L^{p(\\cdot)}(\\mathbb{R})$. We prove that if the operator $aP+Q$ with $P=(I+S)/2$ and $Q=(I-S)/2$ is Fredholm on the variable Lebesgue space $L_N^{p(\\cdot)}(\\mathbb{R})$, then the operators $a_lP+Q$ and $a_rP+Q$ are invertible on standard Lebesgue spaces $L_N^{q_l}(\\mathbb{R})$ and $L_N^"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1105.0407","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-05-02T19:09:32Z","cross_cats_sorted":[],"title_canon_sha256":"cd37e2c7180aa9fabc7dbf91a1259c4f2dfbb964172fd989e654213c38845c56","abstract_canon_sha256":"14875c15433de0c434a5e50a9b9dbb058c0118268cf9edbb741d06a4a772b482"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:20:49.465425Z","signature_b64":"znwjk830SYYKTLFTXEqq7MDd+DqrmU6AUhjd39tQwQ+1OqVy4xZ5QLxLxKWpu3bSCS8+3i/Wq+04XfWAbK1nBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bfced05b44329ef34b1304571c42dd93b971344c5f8d32cf0430787c5e2e6a7e","last_reissued_at":"2026-05-18T04:20:49.464711Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:20:49.464711Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On singular integral operators with semi-almost periodic coefficients on variable Lebesgue spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Alexei Yu. Karlovich, Ilya M. Spitkovsky","submitted_at":"2011-05-02T19:09:32Z","abstract_excerpt":"Let $a$ be a semi-almost periodic matrix function with the almost periodic representatives $a_l$ and $a_r$ at $-\\infty$ and $+\\infty$, respectively. Suppose $p:\\mathbb{R}\\to(1,\\infty)$ is a slowly oscillating exponent such that the Cauchy singular integral operator $S$ is bounded on the variable Lebesgue space $L^{p(\\cdot)}(\\mathbb{R})$. We prove that if the operator $aP+Q$ with $P=(I+S)/2$ and $Q=(I-S)/2$ is Fredholm on the variable Lebesgue space $L_N^{p(\\cdot)}(\\mathbb{R})$, then the operators $a_lP+Q$ and $a_rP+Q$ are invertible on standard Lebesgue spaces $L_N^{q_l}(\\mathbb{R})$ and $L_N^"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1105.0407","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1105.0407","created_at":"2026-05-18T04:20:49.464818+00:00"},{"alias_kind":"arxiv_version","alias_value":"1105.0407v2","created_at":"2026-05-18T04:20:49.464818+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1105.0407","created_at":"2026-05-18T04:20:49.464818+00:00"},{"alias_kind":"pith_short_12","alias_value":"X7HNAW2EGKPP","created_at":"2026-05-18T12:26:44.992195+00:00"},{"alias_kind":"pith_short_16","alias_value":"X7HNAW2EGKPPGSYT","created_at":"2026-05-18T12:26:44.992195+00:00"},{"alias_kind":"pith_short_8","alias_value":"X7HNAW2E","created_at":"2026-05-18T12:26:44.992195+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/X7HNAW2EGKPPGSYTARLRYQW5SO","json":"https://pith.science/pith/X7HNAW2EGKPPGSYTARLRYQW5SO.json","graph_json":"https://pith.science/api/pith-number/X7HNAW2EGKPPGSYTARLRYQW5SO/graph.json","events_json":"https://pith.science/api/pith-number/X7HNAW2EGKPPGSYTARLRYQW5SO/events.json","paper":"https://pith.science/paper/X7HNAW2E"},"agent_actions":{"view_html":"https://pith.science/pith/X7HNAW2EGKPPGSYTARLRYQW5SO","download_json":"https://pith.science/pith/X7HNAW2EGKPPGSYTARLRYQW5SO.json","view_paper":"https://pith.science/paper/X7HNAW2E","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1105.0407&json=true","fetch_graph":"https://pith.science/api/pith-number/X7HNAW2EGKPPGSYTARLRYQW5SO/graph.json","fetch_events":"https://pith.science/api/pith-number/X7HNAW2EGKPPGSYTARLRYQW5SO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/X7HNAW2EGKPPGSYTARLRYQW5SO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/X7HNAW2EGKPPGSYTARLRYQW5SO/action/storage_attestation","attest_author":"https://pith.science/pith/X7HNAW2EGKPPGSYTARLRYQW5SO/action/author_attestation","sign_citation":"https://pith.science/pith/X7HNAW2EGKPPGSYTARLRYQW5SO/action/citation_signature","submit_replication":"https://pith.science/pith/X7HNAW2EGKPPGSYTARLRYQW5SO/action/replication_record"}},"created_at":"2026-05-18T04:20:49.464818+00:00","updated_at":"2026-05-18T04:20:49.464818+00:00"}