The Two-Weight Inequality for the Poisson Operator in the Bessel Setting

Fix $\lambda >0$. Consider the Bessel operator ${\mathrm{\Delta }}_{\lambda }:=-\frac{d^2}{d{x}^2}-\frac{2\lambda }{x}\frac{d}{dx}$ on ${\mathbb{R}}_{+}:=(0,\mathrm{\infty })$ and the harmonic conjugacy introduced by Muckenhoupt and Stein. We provide the two-weight inequality for the Poisson operator ${\mathsf{P}}_t^{[\lambda ]}=e^{-t\sqrt{{\mathrm{\Delta }}_{\lambda }}}$ in this Bessel setting. In particular, we prove that for a measure $\mu$ on ${\mathbb{R}}_{+,+}^2:=(0,\mathrm{\infty })\times (0,\mathrm{\infty })$ and $\sigma$ on ${\mathbb{R}}_{+}$: $$\mathrm{\parallel }{\mathsf{P}}_{\sigma }^{[\lambda ]}(f){\mathrm{\parallel }}_{L^2({\mathbb{R}}_{+,+}^2;\mu )}\lesssim \mathrm{\parallel }f{\mathrm{\parallel }}_{L^2({\mathbb{R}}_{+};\sigma )},$$ if and only if testing conditions hold for the the Poisson operator and its adjoint. Further, the norm of the operator is shown to be equivalent to the best constant in the testing conditions.