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this paper we revisit nonnegative kernels in the first Heisenberg group $\\He$, and in particular we further study the family $$K_\\alpha(x,y,z)= \\frac{|z|^{\\alpha/2}}{\\|(x,y,z)\\|_{H}^{\\alpha+1}}, \\quad \\alpha>0,$$ which was introduced in \\cite{CL}.\n  We first show that if $E \\subset \\He$ is a $1$-Ahlfors regular set and the SIO associated with the kernel $K_4$ is $L^2(E)$-bounded, then $E$ is contained in a $1$-Ahlfors regular curve. Combined with the converse implication which was obtained by F\\\"assler and Orponen in \\cite{FO1dim}, our result provides a characterization of uniform $1$-recti","authors_text":"Lingxiao Zhang, Sean Li, Vasileios Chousionis","cross_cats":["math.MG"],"headline":"L^2 boundedness of the K_4 singular integral on a 1-Ahlfors regular set in the Heisenberg group implies it lies in a 1-Ahlfors regular curve.","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2026-05-17T22:41:12Z","title":"On singular integrals with non-negative kernels in the Heisenberg group"},"references":{"count":22,"internal_anchors":0,"resolved_work":22,"sample":[{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":1,"title":"V . Chousionis, J. Mateu, L. Prat, and X. Tolsa. Calderón-Zygmund kernels and rectifiability in the plane.Adv. Math., 231(1):535–568, 2012","work_id":"caac5554-57bb-4442-a905-405751c9c0ce","year":2012},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":2,"title":"Nonnegative kernels and 1-rectifiability in the Heisenberg group.Anal","work_id":"954215f6-00eb-44ae-a4a0-bb26d1d6dfa7","year":2017},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":3,"title":"Singular integrals onC 1,α intrinsic graphs in step 2 Carnot groups.J","work_id":"63c47907-e998-4235-8d09-3bd7662491d5","year":2025},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":4,"title":"Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1990","work_id":"128ef251-16b4-4b81-be01-403bd6b9f6b2","year":1990},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":5,"title":"A new family of singular integral operators whoseL 2-boundedness implies rectifiability.J","work_id":"1a847e2a-c3a5-4609-ac91-2f76a5611e3f","year":2017}],"snapshot_sha256":"d3919b6551061adfe2656451f764d0745933ee7d8b5fd63b6cde55733fa16904"},"source":{"id":"2605.17680","kind":"arxiv","version":1},"verdict":{"created_at":"2026-05-19T21:57:23.462304Z","id":"ea6d1be5-9ffa-43df-9bb4-21f91a5bfaeb","model_set":{"reader":"grok-4.3"},"one_line_summary":"L2-boundedness of the SIO with kernel K_4 on 1-Ahlfors regular sets in the Heisenberg group characterizes containment in 1-Ahlfors regular curves, with negative results for alpha in (0,2) and a bounded operator on a purely 1-unrectifiable set.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"L^2 boundedness of the K_4 singular integral on a 1-Ahlfors regular set in the Heisenberg group implies it lies in a 1-Ahlfors regular curve.","strongest_claim":"If E subset He is a 1-Ahlfors regular set and the singular integral operator associated with the kernel K_4 is L^2(E)-bounded, then E is contained in a 1-Ahlfors regular curve.","weakest_assumption":"The 1-Ahlfors regularity of the set E together with the non-negativity and precise homogeneity of the kernel K_4 are assumed to control the maximal function and cancellation properties needed for the implication to hold."}},"verdict_id":"ea6d1be5-9ffa-43df-9bb4-21f91a5bfaeb"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:de87ab9dc5317d4fde24db334f74582054f8c45c7b9e1d20366255ea22ebc862","target":"record","created_at":"2026-05-20T00:04:52Z","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":"f20c870ef6d46c6293f68e03f6c40050090fcb08fa56aca1d3142d19765cf305","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2026-05-17T22:41:12Z","title_canon_sha256":"030dd550635b01938a8a6333c504a8839bf2da42b813b623a42fc7d42bd1b714"},"schema_version":"1.0","source":{"id":"2605.17680","kind":"arxiv","version":1}},"canonical_sha256":"c73635eabbeca8bb2c09e4a7bec885907ae1f73860d413c29ab9e17523165f37","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c73635eabbeca8bb2c09e4a7bec885907ae1f73860d413c29ab9e17523165f37","first_computed_at":"2026-05-20T00:04:52.427713Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-20T00:04:52.427713Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ZaDvS2RopuODALPp/3T+e9GewyqXWFeuLh1WxJFcqJtDsj0s0je1/9S18ERuMpe4oYgOTthni6dWRZ9FtPZ4BQ==","signature_status":"signed_v1","signed_at":"2026-05-20T00:04:52.428590Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.17680","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:de87ab9dc5317d4fde24db334f74582054f8c45c7b9e1d20366255ea22ebc862","sha256:c18b2cf591a67fc98f6ad570b2fab5c3ecf7a53a6150c8547bfbaae58881a892"],"state_sha256":"56ee2ca5d92e448e91d535651b46ff0d2c112ab69e7f19244846edeabf23f8f8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XwdHARR7QTaEBZQ5KRdwODam0SSAjFd3wpPBlJ58m0LVbotrvBx71VGEaU+EVBoJmT2//OC8W50EYeEdD9MoDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T06:28:04.469211Z","bundle_sha256":"edca172428dd1e74625de9985936c3939e83efbbbe6a3ed5c85ebb98c2d02023"}}