Abstract
We establish Pitt’s inequality and deduce Beckner’s logarithmic uncertainty principle for the Dunkl transform on \({\mathbb{R}}\). Also, we prove Stein–Weiss inequality for the Dunkl–Bessel potentials.
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Beckner W.: Pitt’s inequality and the uncertainty principle. Proc. Amer. Math. Soc. 123, 1897–1905 (1995)
Beckner W.: Pitt’s inequality with sharp convolution estimates. Proc. Amer. Math. Soc. 136, 1871–1885 (2008)
Betancor J. J., Rodriguez-Mesa L.: On Hankel transformation, convolution operators and multipliers on Hardy type spaces. J. Math. Soc. Japan 53, 687–709 (2001)
Dunkl C. F.: Integral kernels with reflection group invariance. Canad. J. Math. 43, 1213–1227 (1991)
Dunkl C. F.: Hankel transforms associated to finite reflection groups. Contemp. Math. 138, 123–138 (1992)
de Jeu M.F.E.: The Dunkl transform. Invent. Math. 113, 147–162 (1993)
Hassani S., Mustapha S., Sifi M.: Riesz potentials and fractional maximal function for the Dunkl transform. J. Lie Theory 19, 725–734 (2009)
Kawazoe T., Mejjaoli H.: Uncertainty principles for the Dunkl transform. Hiroshima Math. J. 40, 241–268 (2010)
Omri S.: Logarithmic uncertainty principle for the Hankel tansform. Int. Trans. Spec. Funct. 22, 655–670 (2011)
M. Rösler, Bessel-type signed hypergroups on \({\mathbb{R}}\) , in: Probability Measures on Groups and Related Structures, XI (Oberwolfach, 1994), Editors H. Heyer and A. Mukherjea (Oberwolfach, 1994), World Sci. Publ., River Edge, NJ (1995), pp. 292–304.
Rösler M.: An uncertainty principle for the Dunkl transform. Bull. Austral. Math. Soc. 59, 353–360 (1999)
Shimeno N.: A note on the uncertainty principle for the Dunkl transform. J. Math. Sci. Univ. Tokyo 8, 33–42 (2001)
Soltani F.: L p-Fourier multipliers for the Dunkl operator on the real line. J. Funct. Anal. 209, 16–35 (2004)
Soltani F.: Sonine transform associated to the Dunkl kernel on the real line. SIGMA 4, 1–14 (2008)
Soltani F.: A general form of Heisenberg–Pauli–Weyl uncertainty inequality for the Dunkl transform. Int. Trans. Spec. Funct. 24, 401–409 (2013)
K. Stempak, La théorie de Littlewood–Paley pour la transformation de Fourier–Bessel, C.R. Acad. Sci. Paris, 303 Serie I (1) (1986), 15–18.
Thangavelu S., Xu Y.: Convolution operator and maximal function for the Dunkl transform. J. Anal. Math. 97, 25–56 (2005)
Thangavelu S., Xu Y.: Riesz transform and Riesz potentials for Dunkl transform. J. Comput. Appl. Math. 199, 181–195 (2007)
G. N. Watson, A Treatise on Theory of Bessel Functions, Cambridge University Press (Cambridge, 1966).
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Author partially supported by DGRST project 04/UR/15-02 and CMCU program 10G 1503.
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Soltani, F. Pitt’s Inequality and Logarithmic Uncertainty Principle for the Dunkl Transform on \({\mathbb{R}}\) . Acta Math. Hungar. 143, 480–490 (2014). https://doi.org/10.1007/s10474-014-0415-3
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DOI: https://doi.org/10.1007/s10474-014-0415-3
Key words and phrases
- Dunkl transform
- Pitt’s inequality
- logarithmic uncertainty principle
- Dunkl–Bessel potential
- Stein–Weiss inequality