{"paper":{"title":"The Variation of the Fractional Maximal Function of a Radial Function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Hannes Luiro, Jos\\'e Madrid","submitted_at":"2017-10-19T16:26:39Z","abstract_excerpt":"In this paper we study the regularity of the non-centered fractional maximal operator $M_{\\beta}$. As the main result, we prove that there exists $C(n,\\beta)$ such that if $q=n/(n-\\beta)$ and $f$ is a radial function, then $\\|DM_{\\beta}f\\|_{L^{q}(\\mathbb{R}^n)}\\leq C(n,\\beta)\\|Df\\|_{L^{1}(\\mathbb{R}^n)}$. The corresponding result was previously known only if $n=1$ or $\\beta=0$. Our proofs are almost free from one-dimensional arguments. Therefore, we believe that the new approach may be very useful when trying to extend the result for all $f\\in W^{1,1}(\\mathbb{R}^n)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.07233","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"}