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

2017/07/24 by Ji Li, Li, Ji, Brett D. Wick +1
Mathematics · #Advanced Harmonic Analysis Research #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Spectral Theory in Mathematical Physics #math.AP

paper · pdf · doi:10.48550/arxiv.1707.07492

openalex publication_date 2017/07/24 · arxiv created 2019/02/26 · arxiv updated 2019/02/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Fix λ>0. Consider the Bessel operator Δ_λ:=-(d²)/(dx²)-(2λ)/(x)\frac ddx on ℝ₊:=(0,∞) and the harmonic conjugacy introduced by Muckenhoupt and Stein. We provide the two-weight inequality for the Poisson operator P^[λ]_t=e^-t√(Δ_λ) in this Bessel setting. In particular, we prove that for a measure μ on ℝ²_+,+:=(0,∞)× (0,∞) and σ on ℝ₊: ‖P^[λ]_σ(f)‖_L²(ℝ²_+,+;μ) \lesssim ‖f‖_L²(ℝ+;σ), 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.

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