{"paper":{"title":"A non-homogeneous local $Tb$ theorem for Littlewood-Paley $g_{\\lambda}^{*}$-function with $L^p$-testing condition","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.CA","authors_text":"Mingming Cao, Qingying Xue","submitted_at":"2015-07-19T14:22:42Z","abstract_excerpt":"In this paper, we present a local $Tb$ theorem for the non-homogeneous Littlewood-Paley $g_{\\lambda}^{*}$-function with non-convolution type kernels and upper power bound measure $\\mu$. We show that, under the assumptions $\\supp b_Q \\subset Q$, $|\\int_Q b_Q d\\mu| \\gtrsim \\mu(Q)$ and $||b_Q||^p_{L^p(\\mu)} \\lesssim \\mu(Q)$, the norm inequality $\\big\\| g_{\\lambda}^{*}(f) \\big\\|_{L^p(\\mu)} \\lesssim \\big\\| f \\big\\|_{L^p(\\mu)}$ holds if and only if the following testing condition holds : $$\\sup_{Q : cubes \\ in \\ \\Rn} \\frac{1}{\\mu(Q)}\\int_Q \\bigg(\\int_{0}^{\\ell(Q)} \\int_{\\Rn} \\Big(\\frac{t}{t+|x-y|}\\B"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.05291","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"}