{"paper":{"title":"Sharp $L^p$ estimates for discrete second order {R}iesz transforms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Komla Domelevo, Stefanie Petermichl","submitted_at":"2015-07-14T10:35:33Z","abstract_excerpt":"We show that multipliers of second order Riesz transforms on products of discrete abelian groups enjoy the $L^{p} $ estimate $p^{\\ast} -1$, where $p^{\\ast} = \\max \\{ p,q \\}$ and $p$ and $q$ are conjugate exponents. This estimate is sharp if one considers all multipliers of the form $\\sum_i \\sigma_{i} R_{i} R^{\\ast}_{i}$ with $| \\sigma_{i} | \\leqslant 1$ and infinite groups. In the real valued case, we obtain better sharp estimates for some specific multipliers, such as $\\sum_{i} \\sigma_{i} R_{i} R^{\\ast}_{i}$ with $0 \\leqslant \\sigma_{i} \\leqslant 1$. These are the first known precise $L^{p} $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.03796","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"}