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Spaces of ${\mathcal D}_{L^p}-$type and the Hankel convolution


Authors: J. J. Betancor and B. J. González
Journal: Proc. Amer. Math. Soc. 129 (2001), 219-228
MSC (2000): Primary 46F12
DOI: https://doi.org/10.1090/S0002-9939-00-05583-0
Published electronically: August 17, 2000
MathSciNet review: 1707136
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Abstract: In this paper we introduce new function spaces that are denoted by ${\mathcal H}_{\mu ,p}$, $\mu >-1/2$ and $1\leq p\leq \infty ,$ and that are spaces of ${\mathcal D}_{L^{p}}-$type where the Hankel convolution and the Hankel transformation are defined. The spaces ${\mathcal H}_{\mu ,p}$ will play the same role in the Hankel setting that the spaces ${\mathcal D}_{L^{p}}$play in the theory of Fourier transformation.


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  • 1. G. ALTENBURG, Bessel-Transformationen in Raümen von Grundfunktionen über dem intervall $\Omega =(0,\infty )$ und derem Dual-raümen, Math. Nachr. 108 (1982), 197-208.
  • 2. S. ABDULLAH, On convolution operators and multipliers of distributions of $L^{p}-$growth, J. Math. Anal. Appl. 183 1994, 196-207. MR 95c:46060
  • 3. S. ABDULLAH AND S. PILIPOVIC, Bounded subsets in spaces of distributions of $L_{p}-$growth, Hokkaido Math. J. 23 (1994), 51-54. MR 94m:46065
  • 4. J. BARROS-NETO, An introduction to the theory of distributions. Pure and Applied Mathematics, 14. Marcel Dekker, Inc. New York, 1973. MR 57:1113
  • 5. J. J. BETANCOR AND I. MARRERO, Multipliers of Hankel transformable generalized functions, Comment. Math. Univ. Carolin. 33 (1992), 389-401. MR 94f:46051
  • 6. J. J. BETANCOR AND I. MARRERO, The Hankel convolution and the Zemanian spaces $\beta _{\mu }$ and ${\beta }_{\mu }^{\prime }$, Math. Nachr. 160 (1993), 277-298. MR 95j:46042
  • 7. J. J. BETANCOR AND I. MARRERO, Structure and convergence in certain spaces of distributions and the generalized Hankel convolution, Math. Japon. 38 (1993), 1141-1155. MR 95j:46043
  • 8. J. J. BETANCOR AND I. MARRERO, On the topology of the space of Hankel convolution operators, J. Math. Anal. Appl. 201 (1996), 994-1001. MR 97h:46065
  • 9. J. J. BETANCOR AND L. RODR´IGUEZ-MESA, Hankel convolution on distributions spaces with exponential growth, Studia Math. 121 no. 1 (1996), 35-52. MR 98e:46047
  • 10. J. J. BETANCOR AND L. RODR´IGUEZ-MESA, On Hankel convolution equations in distribution spaces, Rocky Mountain J. Math. 29 (1999), 93-114. MR 2000b:46066
  • 11. H. BREZIS, Analyse fonctionnelle. Théorie et applications. Collection Mathématiques Appliquées pour la Maîtrise. Masson, Paris, 1983. MR 85a:46001
  • 12. D. T. HAIMO, Integral equations associated with Hankel convolutions, Trans. Amer. Math. Soc. 116 (1965), 330-375. MR 32:2847
  • 13. C. HERZ, On the mean inversion of Fourier and Hankel transforms, Proc. Nat. Acad. Sci. U. S. A. 40 (1954). 996-999. MR 16:127b
  • 14. I. I. HIRSCHMAN, JR., Variation diminishing Hankel transform, J. Analyse Math. 8 (1960/61), 307-336. MR 28:433
  • 15. D. KOVACEVIC, On the hypoellipticity of convolution equations in the ultradistributions spaces of $L^{q}-$growth, Proc. Amer. Math. Soc. 120 (1994), no. 4, 1181-1190. MR 94f:46050
  • 16. I. MARRERO AND J. J. BETANCOR Hankel convolution of generalized functions, Rend. Mat. Appl. (7), 15 (1995), 351-380. MR 96m:46072
  • 17. D. H. PAHK, On the convolution equations in the spaces of distributions of $L^{p}-$growth, Proc. Amer. Math. Soc. 94 (1985), no. 1, 81-86. MR 86j:46037
  • 18. S. PILIPOVIC, Characterizations of bounded sets in spaces of ultradistributions, Proc. Amer. Math. Soc. 120 (1994), no. 4, 1191-1206. MR 94f:46048
  • 19. S. PILIPOVIC, Elements of ${\mathcal D} _{L^{s}}^{\prime \left( M_{p}\right) }$and ${\mathcal D}_{L^{s}}^{\prime \left\{ M_{p}\right\} }$ as boundary values of holomorphic functions, J. Math. Anal. Appl. 203 (1996), 719-738. MR 97m:46070
  • 20. A.L. SCHWARTZ, An inversion theorem for Hankel transform, Proc. Amer. Math. Soc. 22 (1969), 713-717. MR 39:4616
  • 21. L. SCHWARTZ, Theory of distributions I/II, Hermann, Paris, 1957/1959.
  • 22. B. STANKOVIC, The asymptotic of elements belonging to ${\mathcal D}_{L^{p}}$ and ${\mathcal D}_{L^{p}}^{\prime }$, Univ. u Novom Sadu Zb. Rad. Prirod.-Mat. Fak. Ser. Mat. 18 (1988), no. 1, 169-175. MR 90m:46067
  • 23. K. STEMPAK, La theorie de Littlewood-Paley pour la transformation de Fourier-Bessel, C.R. Acad. Sci. Paris Sér. I Math. 303 (1986), 15-18. MR 87k:42013
  • 24. A. H. ZEMANIAN, Generalized Integral Transformations, Interscience, New York, 1968.

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Additional Information

J. J. Betancor
Affiliation: Departamento de Análisis Matemático, Universidad de La Laguna, 38271 La Laguna, Islas Canarias, Spain
Email: jbetanco@ull.es

B. J. González
Affiliation: Departamento de Análisis Matemático, Universidad de La Laguna, 38271 La Laguna, Islas Canarias, Spain

DOI: https://doi.org/10.1090/S0002-9939-00-05583-0
Keywords: ${\mathcal D}_{L^p}-$spaces, Hankel convolution, Bessel operator
Received by editor(s): January 16, 1998
Received by editor(s) in revised form: April 6, 1999
Published electronically: August 17, 2000
Communicated by: David R. Larson
Article copyright: © Copyright 2000 American Mathematical Society

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