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A generalization of the Hankel transform and the Lorentz multipliers. (English) Zbl 1135.42307

Summary: 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 \((\tilde{T}_{\epsilon}f)^{\wedge}(n)=\phi_{\epsilon}(n)\hat{f}(n)\;(n\in{\mathbb{Z}})\), where \(\hat{f}(n)\) is the coefficient of Jacobi expanstion of \(f\), and \({\mathcal H}_{\alpha}(Tf)(\xi)\) be defined by \(\phi(\xi){\mathcal H}_{\alpha}f(\xi)\;(\xi\in(0,\infty))\), where \({\mathcal H}_{\alpha}f\) is the modified Hankel transform of \(f\) with order \(\alpha\). Then the author [Sci. Math. Jpn. 59, No. 3, 479–488 (2004; Zbl 1071.44005)] 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 [loc. cit.].

MSC:

42A45 Multipliers in one variable harmonic analysis
42C20 Other transformations of harmonic type

Citations:

Zbl 1071.44005
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References:

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