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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 ","authors_text":"Brett D. 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