Tokyo Journal of Mathematics

A Generalization of the Hankel Transform and the Lorentz Multipliers

Enji Sato

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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].

Article information

Tokyo J. Math., Volume 29, Number 1 (2006), 147-166.

First available in Project Euclid: 20 December 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 42C10: Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
Secondary: 42A45: Multipliers 42C20: Other transformations of harmonic type


Sato, Enji. A Generalization of the Hankel Transform and the Lorentz Multipliers. Tokyo J. Math. 29 (2006), no. 1, 147--166. doi:10.3836/tjm/1166661872.

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  • R.Askey, A transplantation theorem for Jacobi series, and Illinois J. of Math., 13 (1969), 589–590.
  • J. J. Betanor and K. Stempak, Relating multipliers and transplantation for Fourier-Bessel expansion and Hankel transform, Tôhoku Math. J., 33 (2001), 109–129.
  • W. C. Connett and A. L. Schwartz, Weak type multipliers for Hankel transforms, Pacific J. of Math., 63 (1976), 125–129.
  • R. Hunt, On L (p,q) spaces, Enseign. Math., 12 (1966), 249–276.
  • S. Igari, On the multipliers of Hankel transform,Tôhoku Math. J., 24 (1972), 201–206.
  • Y. Kanjin, Convergence and divergence almost everywhere of spherical means for radial functions, Proc. Amer. Math. Soc., 103 (1988), 1063–1069.
  • E. Sato, Lorentz multipliers for Hankel transforms, Scienticae Mathematicae Japonicae, 59 (3) (2004), 479–488.
  • K. Stempak, On connections between Hankel, Laguerre and Jacobi transplantations, Tôhoku Math. J., 54 (2002), 471–493.
  • G. Szegö, Orthogonal Polynomials, Amer. Math. Soc. Colloquim, (1959).