{"paper":{"title":"$\\ell^p$-improving inequalities for Discrete Spherical Averages","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Michael T. Lacey, Robert Kesler","submitted_at":"2018-04-26T01:10:28Z","abstract_excerpt":"Let $ \\lambda ^2 \\in \\mathbb N $, and in dimensions $ d\\geq 5$, let $ A_{\\lambda } f (x)$ denote the average of $ f \\;:\\; \\mathbb Z ^{d} \\to \\mathbb R $ over the lattice points on the sphere of radius $\\lambda$ centered at $x$. We prove $ \\ell ^{p}$ improving properties of $ A_{\\lambda }$. \\begin{equation*} \\lVert A_{\\lambda }\\rVert_{\\ell ^{p} \\to \\ell ^{p'}} \\leq C_{d,p, \\omega (\\lambda ^2 )} \\lambda ^{d ( 1-\\frac{2}p)}, \\qquad \\tfrac{d-1}{d+1} < p \\leq \\frac{d} {d-2}. \\end{equation*} It holds in dimension $ d =4$ for odd $ \\lambda ^2 $. The dependence is in terms of $ \\omega (\\lambda ^2 )$, "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.09845","kind":"arxiv","version":11},"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/1804.09845/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"}