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Here we deal with bounded kernels $J$ but with polynomial tails, that is, we assume a lower bound of the form $J(x,y) \\geq c_1|x-y|^{-(n + 2\\sigma)}$, for  $|x - y| > c_2$. Our estimates takes the form $\\|u(t)\\|_{L^q(\\R^n)} \\leq C t^{-\\frac{n}{2\\sigma} (1 - \\frac{1}{q})}$ for $t$ large."},"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":"1307.3372","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2013-07-12T08:38:43Z","cross_cats_sorted":[],"title_canon_sha256":"8191a8db8e14de28446f5861aeb6a70eda8e6ba87e3f6862b9e6c76a37a81992","abstract_canon_sha256":"d4944c4b1aa5d4d7abaa8da3a40f0c2aaf31c0a4c3ed13cb24082403112742ee"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:18:39.305584Z","signature_b64":"FnyVRQrzufCzWF5kbdWuMB3nWvH3M+Bac0TNs5BnFZ7DIOfAm3XFh86f4UPtknWZS20qOSSpfSx7RDXJL8u+CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c32653f937b3ae5abcdcff1ffe4ddb37c28cb7b8e7d1ec10672dde5b37b3e48b","last_reissued_at":"2026-05-18T03:18:39.305035Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:18:39.305035Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Fractional decay bounds for nonlocal zero order heat equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Emmanuel Chasseigne (LMPT, Erwin Topp (LMPT), FRDP), J. 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