{"paper":{"title":"The Two-Weight Inequality for the Poisson Operator in the Bessel Setting","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Brett D. Wick, Ji Li","submitted_at":"2017-07-24T11:25:20Z","abstract_excerpt":"Fix $\\lambda>0$. Consider the Bessel operator $\\Delta_\\lambda:=-\\frac{d^2}{dx^2}-\\frac{2\\lambda}{x}\\frac d{dx}$ on $\\mathbb{R}_+:=(0,\\infty)$ and the harmonic conjugacy introduced by Muckenhoupt and Stein. We provide the two-weight inequality for the Poisson operator $\\mathsf{P}^{[\\lambda]}_t=e^{-t\\sqrt{\\Delta_\\lambda}}$ in this Bessel setting. In particular, we prove that for a measure $\\mu$ on $\\mathbb{R}^2_{+,+}:=(0,\\infty)\\times (0,\\infty)$ and $\\sigma$ on $\\mathbb{R}_+$: $$ \\|\\mathsf{P}^{[\\lambda]}_\\sigma(f)\\|_{L^2(\\mathbb{R}^2_{+,+};\\mu)} \\lesssim \\|f\\|_{L^2(\\mathbb{R}_+;\\sigma)}, $$ if "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.07492","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"}