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K(r,s, z,y)| ~dz\\right]^2 dr <\\infty. $$\n  We prove that the stochastic singular integral of the type $$ \\mathbb{T} g(t,x) :=\\int_0^{t} \\int_{R^d} K(t,s,x,y) g(s,y)dy dW_s $$ is a bounded operator on $\\mathbb{L}_p=L_p(\\Omega \\times (0,\\infty"},"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":"1608.08728","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.PR","submitted_at":"2016-08-31T05:09:04Z","cross_cats_sorted":[],"title_canon_sha256":"d5335e10a1be0b9c464d91e6086b41903c77bf688f63ec7cd9295a96e3ded8fa","abstract_canon_sha256":"d50fac0b0762b943806da275efa89d4d37fc9b9675d935e2ae9836756878c8a3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:42:46.493981Z","signature_b64":"87LEw8zt8JrNnRRM7p29I8ErCGEquskgpuoZmdbAN+37ct5zd7v9bdDlj/e66r8JP/4LGhH479TIrBjjW0fuDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dbc6c118d715ea4fe83fcb19147b1e9df4b3b13cbebfb39d608882ae4be54ba4","last_reissued_at":"2026-05-18T00:42:46.493377Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:42:46.493377Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"An lp-boundedness of stochastic singular integral operators and its application to spdes","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Ildoo Kim, Kyeonghun Kim","submitted_at":"2016-08-31T05:09:04Z","abstract_excerpt":"In this article we introduce a stochastic counterpart of the H\\\"ormander condtion on the kernel $K(r,t,x,y)$: there exists a pseudo-metric $\\rho$ on $(0,\\infty)\\times R^d$ and a positive constant $C_0$ such that for $X=(t,x), Y=(s,y), Z=(r,z) \\in (0,\\infty) \\times R^d$, $$ \\sup_{X,Y}\\int_{0}^\\infty \\left[ \\int_{\\rho(X,Z) \\geq C_0 \\rho(X,Y)} | K(r,t, z,x) - K(r,s, z,y)| ~dz\\right]^2 dr <\\infty. $$\n  We prove that the stochastic singular integral of the type $$ \\mathbb{T} g(t,x) :=\\int_0^{t} \\int_{R^d} K(t,s,x,y) g(s,y)dy dW_s $$ is a bounded operator on $\\mathbb{L}_p=L_p(\\Omega \\times (0,\\infty"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.08728","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":"1608.08728","created_at":"2026-05-18T00:42:46.493458+00:00"},{"alias_kind":"arxiv_version","alias_value":"1608.08728v2","created_at":"2026-05-18T00:42:46.493458+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.08728","created_at":"2026-05-18T00:42:46.493458+00:00"},{"alias_kind":"pith_short_12","alias_value":"3PDMCGGXCXVE","created_at":"2026-05-18T12:29:55.572404+00:00"},{"alias_kind":"pith_short_16","alias_value":"3PDMCGGXCXVE72B7","created_at":"2026-05-18T12:29:55.572404+00:00"},{"alias_kind":"pith_short_8","alias_value":"3PDMCGGX","created_at":"2026-05-18T12:29:55.572404+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/3PDMCGGXCXVE72B7ZMMRI6Y6TX","json":"https://pith.science/pith/3PDMCGGXCXVE72B7ZMMRI6Y6TX.json","graph_json":"https://pith.science/api/pith-number/3PDMCGGXCXVE72B7ZMMRI6Y6TX/graph.json","events_json":"https://pith.science/api/pith-number/3PDMCGGXCXVE72B7ZMMRI6Y6TX/events.json","paper":"https://pith.science/paper/3PDMCGGX"},"agent_actions":{"view_html":"https://pith.science/pith/3PDMCGGXCXVE72B7ZMMRI6Y6TX","download_json":"https://pith.science/pith/3PDMCGGXCXVE72B7ZMMRI6Y6TX.json","view_paper":"https://pith.science/paper/3PDMCGGX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1608.08728&json=true","fetch_graph":"https://pith.science/api/pith-number/3PDMCGGXCXVE72B7ZMMRI6Y6TX/graph.json","fetch_events":"https://pith.science/api/pith-number/3PDMCGGXCXVE72B7ZMMRI6Y6TX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/3PDMCGGXCXVE72B7ZMMRI6Y6TX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/3PDMCGGXCXVE72B7ZMMRI6Y6TX/action/storage_attestation","attest_author":"https://pith.science/pith/3PDMCGGXCXVE72B7ZMMRI6Y6TX/action/author_attestation","sign_citation":"https://pith.science/pith/3PDMCGGXCXVE72B7ZMMRI6Y6TX/action/citation_signature","submit_replication":"https://pith.science/pith/3PDMCGGXCXVE72B7ZMMRI6Y6TX/action/replication_record"}},"created_at":"2026-05-18T00:42:46.493458+00:00","updated_at":"2026-05-18T00:42:46.493458+00:00"}