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Journal of Fourier Analysis and Applications

, Volume 19, Issue 2, pp 417–437 | Cite as

Multivariate Hörmander-Type Multiplier Theorem for the Hankel transform

  • Jacek Dziubański
  • Marcin Preisner
  • Błażej Wróbel
Article

Abstract

Let \(\mathcal{H}(f)(x)=\int_{(0,\infty)^{d}} f(\lambda) E_{x}(\lambda) d\nu(\lambda )\), be the multivariate Hankel transform, where \(E_{x}(\lambda)=\prod_{k=1}^{d} (x_{k} \lambda_{k})^{-\alpha _{k}+1/2}J_{\alpha_{k}-1/2}(x_{k} \lambda_{k})\), with (λ)=λ 2α , α=(α 1,…,α d ). We give sufficient conditions on a bounded function m(λ) which guarantee that the operator \(\mathcal{H}(m\mathcal{H} f)\) is bounded on L p () and of weak-type (1,1), or bounded on the Hardy space H 1((0,∞) d ,) in the sense of Coifman-Weiss.

Keywords

Spectral multiplier Bessel operator Hankel transform Hardy space 

Mathematics Subject Classification

42B15 42B20 42B30 

Notes

Acknowledgements

The authors would like to thank Alessio Martini for discussions on spectral multipliers, Adam Nowak and Tomasz Z. Szarek for their useful remarks, Jacek Zienkiewicz for pointing out to us Example 5.1, and the referees for their helpful comments and suggestions.

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Jacek Dziubański
    • 1
  • Marcin Preisner
    • 1
  • Błażej Wróbel
    • 1
  1. 1.Instytut MatematycznyUniwersytet WrocławskiWrocławPoland

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