{"paper":{"title":"On sharp aperture-weighted estimates for square functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Andrei K. Lerner","submitted_at":"2013-01-06T20:41:33Z","abstract_excerpt":"Let $S_{\\a,\\psi}(f)$ be the square function defined by means of the cone in ${\\mathbb R}^{n+1}_{+}$ of aperture $\\a$, and a standard kernel $\\psi$. Let $[w]_{A_p}$ denote the $A_p$ characteristic of the weight $w$. We show that for any $1<p<\\infty$ and $\\a\\ge 1$, $$\\|S_{\\a,\\psi}\\|_{L^p(w)}\\lesssim \\a^n[w]_{A_p}^{\\max(1/2,\\frac{1}{p-1})}.$$ For each fixed $\\a$ the dependence on $[w]_{A_p}$ is sharp. Also, on all class $A_p$ the result is sharp in $\\a$. Previously this estimate was proved in the case $\\a=1$ using the intrinsic square function. However, that approach does not allow to get the abo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1301.1051","kind":"arxiv","version":2},"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"}