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Seiringer we obtain: 1- The following improved Hardy inequality for $p\\ge 2$ For all $q<p$, there exists a positive constant $C\\equiv C(\\Omega, q, N, s)$ such that\n  $$ \\int_{{\\mathbb R}^N}\\int_{{\\mathbb R}^N} \\, \\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+ps}}\\,dx\\,dy - \\Lambda_{N,p,s} \\int_{{\\mathbb R}^N} \\frac{|u(x)|^p}{|x|^{p}}\\,dx\\geq C \\int_{\\Omega}\\dint_{\\Omega}\\frac{|u(x)-u(y)|^p}{|x-y|^{N+qs}}dxdy $$ for all $u \\in \\mathcal{C}_0^\\in"},"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":"1611.04724","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-11-15T07:36:36Z","cross_cats_sorted":[],"title_canon_sha256":"3ded740fbc4cfba6f19d74e36e16354cff8b7fe50cc3fb9d5d9add8c5b90d495","abstract_canon_sha256":"fb801ab3a78aee5f0347443da1b138edcc0e1abf808b665c80d950a645c87e35"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:58:50.443020Z","signature_b64":"4SPw/4tMCVs9z9OjMsa4xN099BFIO0BQGHjMUNN7fc4ofZKrSV9vaS5MuzrFeGL2mAg2r9bMkNYe0BfjsVzrDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e5e4b6635b979ce95ca1c30a1597ff80355b9ead93580e352a9e6b785964c13b","last_reissued_at":"2026-05-17T23:58:50.442331Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:58:50.442331Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Caffarelli-Kohn-Nirenberg type inequalities of fractional order with applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Boumediene Abdellaoui, Rachid Bentifour","submitted_at":"2016-11-15T07:36:36Z","abstract_excerpt":"Let $0<s<1$ and $p>1$ be such that $ps<N$. Assume that $\\Omega$ is a bounded domain containing the origin. Staring from the ground state inequality by R. Frank and R. Seiringer we obtain: 1- The following improved Hardy inequality for $p\\ge 2$ For all $q<p$, there exists a positive constant $C\\equiv C(\\Omega, q, N, s)$ such that\n  $$ \\int_{{\\mathbb R}^N}\\int_{{\\mathbb R}^N} \\, \\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+ps}}\\,dx\\,dy - \\Lambda_{N,p,s} \\int_{{\\mathbb R}^N} \\frac{|u(x)|^p}{|x|^{p}}\\,dx\\geq C \\int_{\\Omega}\\dint_{\\Omega}\\frac{|u(x)-u(y)|^p}{|x-y|^{N+qs}}dxdy $$ for all $u \\in \\mathcal{C}_0^\\in"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.04724","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":"1611.04724","created_at":"2026-05-17T23:58:50.442439+00:00"},{"alias_kind":"arxiv_version","alias_value":"1611.04724v2","created_at":"2026-05-17T23:58:50.442439+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.04724","created_at":"2026-05-17T23:58:50.442439+00:00"},{"alias_kind":"pith_short_12","alias_value":"4XSLMY23S6OO","created_at":"2026-05-18T12:29:58.707656+00:00"},{"alias_kind":"pith_short_16","alias_value":"4XSLMY23S6OOSXFB","created_at":"2026-05-18T12:29:58.707656+00:00"},{"alias_kind":"pith_short_8","alias_value":"4XSLMY23","created_at":"2026-05-18T12:29:58.707656+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/4XSLMY23S6OOSXFBYMFBLF77QA","json":"https://pith.science/pith/4XSLMY23S6OOSXFBYMFBLF77QA.json","graph_json":"https://pith.science/api/pith-number/4XSLMY23S6OOSXFBYMFBLF77QA/graph.json","events_json":"https://pith.science/api/pith-number/4XSLMY23S6OOSXFBYMFBLF77QA/events.json","paper":"https://pith.science/paper/4XSLMY23"},"agent_actions":{"view_html":"https://pith.science/pith/4XSLMY23S6OOSXFBYMFBLF77QA","download_json":"https://pith.science/pith/4XSLMY23S6OOSXFBYMFBLF77QA.json","view_paper":"https://pith.science/paper/4XSLMY23","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1611.04724&json=true","fetch_graph":"https://pith.science/api/pith-number/4XSLMY23S6OOSXFBYMFBLF77QA/graph.json","fetch_events":"https://pith.science/api/pith-number/4XSLMY23S6OOSXFBYMFBLF77QA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/4XSLMY23S6OOSXFBYMFBLF77QA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/4XSLMY23S6OOSXFBYMFBLF77QA/action/storage_attestation","attest_author":"https://pith.science/pith/4XSLMY23S6OOSXFBYMFBLF77QA/action/author_attestation","sign_citation":"https://pith.science/pith/4XSLMY23S6OOSXFBYMFBLF77QA/action/citation_signature","submit_replication":"https://pith.science/pith/4XSLMY23S6OOSXFBYMFBLF77QA/action/replication_record"}},"created_at":"2026-05-17T23:58:50.442439+00:00","updated_at":"2026-05-17T23:58:50.442439+00:00"}