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Mediterranean Journal of Mathematics

, Volume 1, Issue 3, pp 283–295 | Cite as

An L p -L q -Version of Morgan’s Theorem for the Dunkl-Bessel Transform

  • Hatem Mejjaoli
  • Khalifa Trimèche
Original paper
  • 47 Downloads

Abstract.

In this paper, we give an L p -L q -version of Morgan’s theorem for the Dunkl-Bessel transform \(\mathcal{F}_{D,B} \) on \(\mathbb{R}_ + ^{d + 1} .\) More precisely, we prove that for all \(1 \leq p,q \leq + \infty ,\;\alpha > 2,\;\eta = \frac{\alpha } {{\alpha - 1}}\) and \(a > 0,b > 0,\) then for all measurable function f on \(\mathbb{R}_ + ^{d + 1} ,\) the conditions \(e^{a||x||^\alpha } f \in L_{k,\beta }^p (\mathbb{R}_ + ^{d + 1} )\) and \(e^{b||y||^\eta } \mathcal{F}_{D,B} (f) \in L_{k,\beta }^q (\mathbb{R}_ + ^{d + 1} )\) imply f = 0, if and only if \((a\alpha )^{\frac{1} {\alpha }} (b\eta )^{\frac{1} {\eta }} > (\sin \left( {\frac{\Pi } {2}} \right)(\eta - 1))^{\frac{1} {\eta }} ,\) where \(L_{k,\beta }^p (\mathbb{R}_ + ^{d + 1} ),\) are the Lebesgue spaces associated with the Dunkl-Bessel transform.

Mathematics Subject Classification (2000).

Primary 35R10 Secondary 44A15 

Keywords.

Dunkl-Bessel transform Morgan’s theorem 

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Copyright information

© Birkhäuser Verlag, Basel 2004

Authors and Affiliations

  1. 1.Department of MathematicsFaculty of Sciences of Tunis - CAMPUSTunisTunisia

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