{"paper":{"title":"The Maximal Function and Square Function Control the Variation: An Elementary Proof","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CA","authors_text":"Bartosz Trojan, Ben Krause, Kevin Hughes","submitted_at":"2014-08-06T08:45:48Z","abstract_excerpt":"In this note we prove the following good-$\\lambda$ inequality, for $r>2$, all $\\lambda > 0$, $\\delta \\in \\big(0, \\frac{1}{2} \\big)$ \\[ \\nu\\big\\{ V_r(f) > 3 \\lambda ; \\mathcal{M}(f) \\leq \\delta \\lambda\\big\\} \\leq 4 \\nu\\{s(f) > \\delta \\lambda\\} + {\\delta^2 \\left(1+\\frac{16}{r-2}\\right)^2} \\cdot \\nu\\big\\{ V_r(f) > \\lambda\\big\\}, \\] where $\\mathcal{M}(f)$ is the martingale maximal function, $s(f)$ is the conditional martingale square function. This immediately proves that $V_r(f)$ is bounded on $L^p$, $1 < p <\\infty$ and moreover is integrable when the maximal function is."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.1213","kind":"arxiv","version":3},"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"}