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As shown by Walther (2002, doi:10.1006/jfan.2001.3863), this transform plays an essential role in the study of optimal constants of smoothing estimates for the free Schr\\\"{o}dinger equations on $\\mathbb{R}^d$. On the other hand, Bez et al. (2015, doi:10.1016/j.aim.2015.08.025) studied these optimal constants using a different method, and obtained a certain alternative expression for $T_{\\nu} f$ involving the $"},"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":"2511.00137","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CA","submitted_at":"2025-10-31T15:58:10Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"78b70922e68203d97a232e16000939b1844fd12183e6d384fee5f35cb2b088fd","abstract_canon_sha256":"174a88892d467b06a12b3dac5182d595750a754722bee8997e95e2897570c570"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-03T01:05:06.933353Z","signature_b64":"YvTYBzuZnIRCPn0uJqVq3ZKybXfSqvHBlP9nIEqwMu7H+D9Iiaw/qx9WFZg9JAXyGudPQnSCyFBuFBz7YOSMCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"91b7abaa215ac14170d46c00e423fe4ab7a9ab904455e2260461e34002ee7ac2","last_reissued_at":"2026-06-03T01:05:06.932791Z","signature_status":"signed_v1","first_computed_at":"2026-06-03T01:05:06.932791Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Identities and inequalities for integral transforms involving squares of the Bessel functions","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.CA","authors_text":"Soichiro Suzuki","submitted_at":"2025-10-31T15:58:10Z","abstract_excerpt":"We consider an integral transform given by $T_{\\nu} f(s) := \\pi \\int_0^\\infty rs J_{\\nu}(r s)^2 f(r) \\, dr$, where $J_{\\nu}$ denotes the Bessel function of the first kind of order $\\nu$. As shown by Walther (2002, doi:10.1006/jfan.2001.3863), this transform plays an essential role in the study of optimal constants of smoothing estimates for the free Schr\\\"{o}dinger equations on $\\mathbb{R}^d$. On the other hand, Bez et al. 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