Uniform weighted inequalities for the Hankel transform transplantation operator

TL;DR

Addresses uniform weighted $L^p$ bounds for the Hankel transform transplantation operator across orders $a+k$ and $b+k$ with $a\\neq b$. The approach uses the hypergeometric kernel representation and an auxiliary integral estimate to obtain uniform Calderón-Zygmund bounds, yielding bounds $\\|S_k^{a,b} f\\|_{L^p(u)} \\le C \\|f\\|_{L^p(u)}$ with $C$ independent of $k$ and $u\\in A_p$. The paper also derives a higher-dimensional transference principle (a Rubio de Francia-type result) in mixed-norm and vector-valued settings for radial multipliers. These results contribute to weighted harmonic analysis, providing uniform kernel control and transfer tools for Hankel/Fourier radial operators under $A_p$ weights.

Abstract

In this paper we present uniform weighted inequalities for the Hankel transform transplantation operator. A weighted vector-valued inequality is also obtained. As a consequence we deduce an extension of a transference theorem due to Rubio de Francia.

Theorems & Definitions (10)

• Theorem 1.1
• Theorem 1.2
• Proposition 2.1
• Lemma 2.2
• proof : Proof of Proposition \ref{['prop:C-Z']}
• proof : Proof of Theorem \ref{['thm:main1']}
• Theorem 3.1
• Proposition 3.2
• proof
• proof : Proof of Theorem \ref{['thm:mixed-norm']}