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

TL;DR

This paper establishes a two-weight inequality for the Poisson operator associated with the Bessel operator, characterizing boundedness via testing conditions and providing a precise norm equivalence.

Contribution

It extends two-weight inequality theory to the Bessel setting for the Poisson operator, including necessary and sufficient testing conditions and norm characterization.

Findings

Proved the two-weight inequality for the Bessel Poisson operator.

Characterized the operator norm via testing conditions.

Established equivalence of the norm to the best constant in testing conditions.

Abstract

Fix $\lambda>0$. 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 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.