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We show that any critical point $f\\in L^{(d+1)/d}$ of the functional $\\norm{Tf}_{d+1}/\\norm{f}_{(d+1)/d}$ is infinitely differentiable, and that $|x|^\\delta f\\in L^{(d+1)/d}$ for some $\\delta>0$. In particular, this holds for all extremizers of the associated inequality. This is done by exploiting a generalized Euler-Lagrange equation, and certain weighted norm inequalities for $T$."},"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":"1012.5458","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2010-12-25T07:35:38Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"57963b6cbdd8b95ff561e7266c617dd273819a43d1a2627b5645562741244bdd","abstract_canon_sha256":"827a22bfd1bf48074391a758d4214cb4d145ce52b6e80a2f5434832b15e50aa6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:32:23.433874Z","signature_b64":"wFAQgfRG+GUkQ7PcO/eMayWN/zxBbAlNZ/Qx0wjJJOuQs0rwU9Nz4lsCwhXkpHDhBodaEwsZ3bbelvhVBiHhDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b320c9bd307566d46c1853e3c6a893e6375aa75393240659ecb23c5852a71dbc","last_reissued_at":"2026-05-18T04:32:23.433128Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:32:23.433128Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Smoothness of Extremizers of a Convolution Inequality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.CA","authors_text":"Michael Christ, Qingying Xue","submitted_at":"2010-12-25T07:35:38Z","abstract_excerpt":"Let $d\\ge 2$ and $T$ be the convolution operator $Tf(x)=\\int_{\\reals^{d-1}} f(x'-t,x_d-|t|^2)\\,dt$, which is is bounded from $L^{(d+1)/d}(\\reals^d)$ to $L^{d+1}(\\reals^d)$. We show that any critical point $f\\in L^{(d+1)/d}$ of the functional $\\norm{Tf}_{d+1}/\\norm{f}_{(d+1)/d}$ is infinitely differentiable, and that $|x|^\\delta f\\in L^{(d+1)/d}$ for some $\\delta>0$. In particular, this holds for all extremizers of the associated inequality. 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