Abstract

Fix λ>0. Consider the Bessel operator Δλ:=−d2dx2−2λxddx on R+:=(0,∞) and the harmonic conjugacy introduced by Muckenhoupt and Stein. We provide the two-weight inequality for the Poisson operator Pt[λ]=e−tΔλ in this Bessel setting. In particular, we prove that for a measure μ on R+,+2:=(0,∞)×(0,∞) and σ on R+:‖Pσ[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.

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