{"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. This is done by exploiting a generalized Euler-Lagrange equation, and certain weighted norm inequalities for $T$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.5458","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}