The two-weight inequality for the Poisson operator in the Bessel setting

Fix λ>0. Consider the Bessel operator [Formula presented] on R₊:=(0,∞) and the harmonic conjugacy introduced by Muckenhoupt and Stein. We provide the two-weight inequality for the Poisson operator P_t ^[λ]=e^−tΔ_λ in this Bessel setting. In particular, we prove that for a measure μ on R_+,+ ²:=(0,∞)×(0,∞) and σ on R₊: ‖P_σ ^[λ](f)‖_{L²(R+,+ ²;μ)}≲‖f‖_L²(R+;σ), if and only if testing conditions hold for 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.