A Generalization of the Hankel Transform and the Lorentz Multipliers

Abstract

Let $\phi$ be a bounded function on $[0,\infty)$ continuous except on a null set, and $\phi_{\epsilon}(\xi)=\phi(\epsilon\xi)\ (\epsilon>0).$ Also let $\tilde{T}_{\epsilon}$ be the operator on Jacobi series such that $(\tilde{T}_{\epsilon}f)^{\wedge}(n)=\phi_{\epsilon}(n)\hat{f}(n)\ (n\in{\bf Z})$, where $\hat{f}(n)$ is the coefficient of Jacobi expanstion of $f$, and ${\cal H}_{\alpha}(Tf)(\xi)$ be defined by $\phi(\xi){\cal H}_{\alpha}f(\xi)\ (\xi\in(0,\infty))$, where ${\cal H}_{\alpha}f$ is the modified Hankel transform of $f$ with order $\alpha$. Then the author [7] proved that if the operator norm of $\tilde{T}_{\epsilon}$ is uniformly bounded for all $\epsilon>0$, $T$ is a bounded operator on the modified Hankel transforms in the Lorentz spaces, and we have the maximal type theorem in the Lorentz spaces, respectively. In this paper, we give a generalized definition of the modified Hankel transform and the Hankel transform, and prove a generalization of the results in [7].