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.].


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


Zbl 1071.44005
Full Text: DOI


[1] R.Askey, A transplantation theorem for Jacobi series, and Illinois J. of Math., 13 (1969), 589-590. · Zbl 0174.35303
[2] J. J. Betanor and K. Stempak, Relating multipliers and transplantation for Fourier-Bessel expansion and Hankel transform, Tôhoku Math. J., 33 (2001), 109-129. · Zbl 0992.42009 · doi:10.2748/tmj/1178207534
[3] W. C. Connett and A. L. Schwartz, Weak type multipliers for Hankel transforms, Pacific J. of Math., 63 (1976), 125-129. · Zbl 0315.44004 · doi:10.2140/pjm.1976.63.125
[4] R. Hunt, On L (p,q) spaces, Enseign. Math., 12 (1966), 249-276. · Zbl 0181.40301
[5] S. Igari, On the multipliers of Hankel transform,Tôhoku Math. J., 24 (1972), 201-206. · Zbl 0239.42004 · doi:10.2748/tmj/1178241530
[6] Y. Kanjin, Convergence and divergence almost everywhere of spherical means for radial functions, Proc. Amer. Math. Soc., 103 (1988), 1063-1069. JSTOR: · Zbl 0671.42016 · doi:10.2307/2047086
[7] E. Sato, Lorentz multipliers for Hankel transforms, Scienticae Mathematicae Japonicae, 59 (3) (2004), 479-488. · Zbl 1071.44005
[8] K. Stempak, On connections between Hankel, Laguerre and Jacobi transplantations, Tôhoku Math. J., 54 (2002), 471-493. · Zbl 1040.42022 · doi:10.2748/tmj/1113247646
[9] G. Szegö, Orthogonal Polynomials, Amer. Math. Soc. Colloquim, (1959).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.