Korean Journal of Mathematics

  • 제22권1호
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  • Pages.1-28
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  • 2014
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  • 1976-8605(pISSN)
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  • 2288-1433(eISSN)
강원경기수학회 (The Kangwon-Kyungki Mathematical Society)
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THE POSITIVITY OF THE HYPERGEOMETRIC TRANSLATION OPERATORS ASSOCIATED TO THE CHEREDNIK OPERATORS AND THE HECKMAN-OPDAM THEORY ATTACHED TO THE ROOT SYSTEMS OF TYPE B2 AND C2

  • Trimeche, Khalifa (Department of Mathematics Faculty of Sciences of Tunis University of Tunis El Manar)
  • 투고 : 2013.02.06
  • 심사 : 2013.03.13
  • 발행 : 2014.03.30
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초록

We consider the hypergeometric translation operator associated to the Cherednik operators and the Heckman-Opdam theory attached to the root system of type $B_2$. We prove in this paper that these operators are positivity preserving and allow positive integral representations. In particular we deduce that the product formulas of the Opdam-Cherednik and the Heckman-Opdam kernels are positive integral transforms, and we obtain best estimates of these kernels. The method used to obtain the previous results shows that these results are also true in the case of the root system of type $C_2$.

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참고문헌

  1. I. Cherednik, A unification of Knizhnik-Zamslodchnikov equations and Dunkl operators via affine Hecke algebras, Invent. Math. 106 (1990), 411-432.
  2. G. J. Heckman and E. M. Opdam , Root systems and hypergeometric functions I, Compos Math. 64 (1987), 329-352.
  3. E. M. Opdam, Harmonic analysis for certain representations of graded Hecke algebras, Acta Math. 175 (1995), 75-121. https://doi.org/10.1007/BF02392487
  4. W. Rudin, Real and complex analysis. Second Edition, McGraw-Hill Book Company, New York, London, Sydney, Tokyo, 1974.
  5. B. Schapira, Contribution to the hypergeometric function theory of Heckman and Opdam : sharp estimates, Schwartz spaces, heat kernel, Geom. Funct. Anal. 18 (2008), 222-250. https://doi.org/10.1007/s00039-008-0658-7
  6. K. Trimeche, The trigonometric Dunkl intertwining operator and its dual associated with the Cherednik operators and the Heckman-Opdam theory, Adv. Pure Appl. Math. 1 (2010), 293-323.
  7. K. Trimeche, Harmonic analysis associated with the Cherednik operators and the Heckam-Opdam theory, Adv. Pure Appl. Math. 2 (2011), 23-46.
  8. K. Trimeche, Hypergeometric convolution structure on $L^p$-spaces and Applications, for the Heckman-Opdam theory, Preprint. Faculty of Sciences of Tunis. 2012.
  9. K. Trimeche, Positivity of the hypergeometric translation operators associated with the Cherednik operator, and of the Dunkl translation operators in the one dimentional case, Preprint. Faculty of Sciences of Tunis. 2013.
  10. K. Trimeche, The positivity of the hypergeometric translation operator and of its dual associated with the Cherednik operators and the Heckman-Opdam theory on ${\mathbb{R}}^d$, Preprint. Faculty of Sciences of Tunis. 2013.
  11. K. Trimeche, The positivity of the transmutation operators associated with the Cherednik operators attached to the root system of type $B_2$ and $C_2$, Preprint. Faculty of Sciences of Tunis. 2014.

피인용 문헌

  1. Qualitative uncertainty principles for the hypergeometric Fourier transform vol.145, pp.1, 2015, https://doi.org/10.1007/s10474-014-0466-5
  2. Evolution of Samsung group and its central office: Imperfect market and capacity-building vol.15, pp.5, 2016, https://doi.org/10.1057/s41291-016-0011-1
  3. On the range of the Opdam–Cherednik transform and Roe’s theorem in the Cherednik setting vol.7, pp.1, 2016, https://doi.org/10.1007/s11868-015-0137-5
  4. ABSOLUTE CONTINUITY OF THE REPRESENTING MEASURES OF THE HYPERGEOMETRIC TRANSLATION OPERATORS ATTACHED TO THE ROOT SYSTEM OF TYPE B2 AND C2 vol.22, pp.4, 2014, https://doi.org/10.11568/kjm.2014.22.4.711
  5. QUALITATIVE UNCERTAINTY PRINCIPLES FOR THE INVERSE OF THE HYPERGEOMETRIC FOURIER TRANSFORM vol.23, pp.1, 2014, https://doi.org/10.11568/kjm.2015.23.1.129