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We consider the operator \\begin{equation*} T_\\theta f(x) = \\lim_{\\varepsilon\\to 0^+} \\int_\\varepsilon^1 e^{i\\gamma(t)}f(x-t) \\frac{dt}{t^{\\theta}\\psi(t)^{1-\\theta}}, \\end{equation*} where $\\gamma$ is a real function with $\\lim_{t\\to 0^+}|\\gamma(t)| = \\infty$ and $0 \\le \\theta \\le 1$. Assuming certain regularity and growth conditions on $\\psi$ and $\\gamma$, we show that $T_1$ is of weak type $(1,1)$."},"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":"1802.04767","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-02-13T17:59:50Z","cross_cats_sorted":[],"title_canon_sha256":"c19e912e455a54943e646becea71412617746d6ad5b124364158ebd45f3cc947","abstract_canon_sha256":"c9f8fd29f6470928b4853999a5683cf7f59404659e898dbc922cab214f046f96"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:56:53.579577Z","signature_b64":"9MqvWGu36vPJrl+VaSelUUnjLX6LBPd2knOPTn+1VfpLZ/z6XntdaPhdxRSK9jLUvgUtpKLOy+ysERcWhLO9AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8b8d920d007f87285aae12b00114a4a046ba03db9616453d7162e922d92cba48","last_reissued_at":"2026-05-17T23:56:53.578961Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:56:53.578961Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Weak-type (1,1) estimates for strongly singular operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Magali Folch-Gabayet, Ricardo A. S\\'aenz","submitted_at":"2018-02-13T17:59:50Z","abstract_excerpt":"Let $\\psi$ be a positive function defined near the origin such that $\\lim_{t\\to 0^{+}}\\psi(t)=0$. We consider the operator \\begin{equation*} T_\\theta f(x) = \\lim_{\\varepsilon\\to 0^+} \\int_\\varepsilon^1 e^{i\\gamma(t)}f(x-t) \\frac{dt}{t^{\\theta}\\psi(t)^{1-\\theta}}, \\end{equation*} where $\\gamma$ is a real function with $\\lim_{t\\to 0^+}|\\gamma(t)| = \\infty$ and $0 \\le \\theta \\le 1$. 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