Hankel multipliers and transplantation operators

Krzysztof Stempak; Walter Trebels

Studia Mathematica (1997)

  • Volume: 126, Issue: 1, page 51-66
  • ISSN: 0039-3223

Abstract

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Connections between Hankel transforms of different order for L p -functions are examined. Well known are the results of Guy [Guy] and Schindler [Sch]. Further relations result from projection formulae for Bessel functions of different order. Consequences for Hankel multipliers are exhibited and implications for radial Fourier multipliers on Euclidean spaces of different dimensions indicated.

How to cite

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Stempak, Krzysztof, and Trebels, Walter. "Hankel multipliers and transplantation operators." Studia Mathematica 126.1 (1997): 51-66. <http://eudml.org/doc/216443>.

@article{Stempak1997,
abstract = {Connections between Hankel transforms of different order for $L^p$-functions are examined. Well known are the results of Guy [Guy] and Schindler [Sch]. Further relations result from projection formulae for Bessel functions of different order. Consequences for Hankel multipliers are exhibited and implications for radial Fourier multipliers on Euclidean spaces of different dimensions indicated.},
author = {Stempak, Krzysztof, Trebels, Walter},
journal = {Studia Mathematica},
keywords = {Hankel transform and multipliers; transplantation; Hankel transforms; Bessel functions; radial Fourier multipliers; Hankel multipliers},
language = {eng},
number = {1},
pages = {51-66},
title = {Hankel multipliers and transplantation operators},
url = {http://eudml.org/doc/216443},
volume = {126},
year = {1997},
}

TY - JOUR
AU - Stempak, Krzysztof
AU - Trebels, Walter
TI - Hankel multipliers and transplantation operators
JO - Studia Mathematica
PY - 1997
VL - 126
IS - 1
SP - 51
EP - 66
AB - Connections between Hankel transforms of different order for $L^p$-functions are examined. Well known are the results of Guy [Guy] and Schindler [Sch]. Further relations result from projection formulae for Bessel functions of different order. Consequences for Hankel multipliers are exhibited and implications for radial Fourier multipliers on Euclidean spaces of different dimensions indicated.
LA - eng
KW - Hankel transform and multipliers; transplantation; Hankel transforms; Bessel functions; radial Fourier multipliers; Hankel multipliers
UR - http://eudml.org/doc/216443
ER -

References

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  1. [CW] R. Coifman and G. Weiss, Some examples of transference methods in harmonic analysis, in: Symposia Math. 22, Academic Press, New York, 1977, 33-45. 
  2. [EMOT] A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Tables of Integral Transforms, Vol. 2, McGraw-Hill, New York, 1954. Zbl0055.36401
  3. [GT] G. Gasper and W. Trebels, Necessary conditions for Hankel multipliers, Indiana Univ. Math. J. 31 (1982), 403-414. Zbl0494.44003
  4. [Guy] D. L. Guy, Hankel multiplier transformations and weighted p-norms, Trans. Amer. Math. Soc. 95 (1960), 137-189. Zbl0091.10202
  5. [He] C. Herz, On the mean inversion of Fourier and Hankel transforms, Proc. Nat. Acad. Sci. U.S.A. 40 (1954), 996-999. Zbl0059.09901
  6. [Hi1] I. I. Hirschman, Jr., Multiplier transformations, II, Duke Math. J. 28 (1961), 45-56. 
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  8. [MWY] B. Muckenhoupt, R. L. Wheeden and W.-S. Young, L 2 multipliers with power weights, Adv. Math. 49 (1983), 170-216. 
  9. [RdF] J. L. Rubio de Francia, Transference principles for radial multipliers, Duke Math. J. 58 (1989), 1-19. 
  10. [SKM] S. G. Samko, A. A. Kilbas and O. I. Marichev, Integrals and Derivatives of Fractional Order and Some of Their Applications, Nauka i Tekhnika, Minsk, 1987 (in Russian). Zbl0617.26004
  11. [Sch] S. Schindler, Explicit integral transform proofs of some transplantation theorems for the Hankel transform, SIAM J. Math. Anal. 4 (1973), 367-384. Zbl0227.44007
  12. [To] A. Torchinsky, Real-Variable Methods in Harmonic Analysis, Academic Press, 1986. 
  13. [T] W. Trebels, Some Fourier multiplier criteria and the spherical Bochner-Riesz kernel, Rev. Roumaine Math. Pures Appl. 20 (1975), 1173-1185. Zbl0328.42003
  14. [W] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Univ. Press, Cambridge, 1966. Zbl0174.36202
  15. [Wi] G. M. Wing, On the L p theory of Hankel transforms, Pacific J. Math. 1 (1951), 313-319. Zbl0044.10906
  16. [Z1] A. H. Zemanian, A distributional Hankel transformation, SIAM J. Appl. Math. 14 (1966), 561-576. Zbl0154.13803
  17. [Z2] A. H. Zemanian, Hankel transform of arbitrary order, Duke Math. J. 34 (1967), 761-769. Zbl0177.39903

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