{"paper":{"title":"Equivalent Characterizations for Boundedness of Maximal Singular Integrals on $ax+b$\\,--Groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.CA","authors_text":"Dachun Yang, Liguang Liu, Maria Vallarino","submitted_at":"2010-07-31T02:17:01Z","abstract_excerpt":"Let $(S, d, \\rho)$ be the affine group $\\mathrm{R}^n \\ltimes \\mathrm{R}^+$ endowed with the left-invariant Riemannian metric $d$ and the right Haar measure $\\rho$, which is of exponential growth at infinity. In this paper, for any linear operator $T$ on $(S, d, \\rho)$ associated with a kernel $K$ satisfying certain integral size condition and H\\\"ormander's condition, the authors prove that the following four statements regarding the corresponding maximal singular integral $T^\\ast$ are equivalent: $T^\\ast$ is bounded from $L_c^\\infty$ to $\\mathrm{BMO}$, $T^\\ast$ is bounded on $L^p$ for all $p\\i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1008.0043","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"}