{"paper":{"title":"Averages along the Square Integers: $\\ell^p$ improving and Sparse Inequalities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Fan Yang, Michael T Lacey, Rui Han","submitted_at":"2019-07-12T13:22:17Z","abstract_excerpt":"Let $f\\in \\ell^2(\\mathbb Z)$. Define the average of $ f$ over the square integers by $ A_N f(x):=\\frac{1}{N}\\sum_{k=1}^N f(x+k^2) $. We show that $ A_N$ satisfies a local scale-free $ \\ell ^{p}$-improving estimate, for $ 3/2 < p \\leq 2$: \\begin{equation*}\n  N ^{-2/p'} \\lVert A_N f \\rVert _{ p'} \\lesssim N ^{-2/p} \\lVert f\\rVert _{\\ell ^{p}}, \\end{equation*} provided $ f$ is supported in some interval of length $ N ^2 $, and $ p' =\\frac{p} {p-1}$ is the conjugate index. The inequality above fails for $ 1< p < 3/2$. The maximal function $ A f = \\sup _{N\\geq 1} |A_Nf| $ satisfies a similar sparse"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.05734","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1907.05734/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}