The Dunkl-Hausdorff operators and the Dunkl continuous wavelets transform
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The Dunkl-Hausdorff operators and the Dunkl continuous wavelets transform Radouan Daher1 · Faouaz Saadi1 Received: 24 January 2020 / Revised: 16 March 2020 / Accepted: 18 June 2020 © Springer Nature Switzerland AG 2020
Abstract In the present paper, we prove the boundedness of Dunkl-Hausdorff operators in p space L α (R) and in the Hardy space Hα1 (R) associated with the Dunkl operators, investigate continuous Dunkl wavelet transformation, and obtain some useful results. The relation between Dunkl wavelet transformation and Dunkl-Hausdorff operator is also established. Keywords Dunkl transformation · Dunkl-Hausdorff operator · Riesz transformation · Hardy space · Dunkl wavelet transformation Mathematics Subject Classification 47G10 · 47B38 · 43A32
1 Introduction The Hausdorff operator is one of the most important operators in harmonic analysis, and it is used to solve certain classical problems in analysis. The modern study of general Hausdorff operators on L 1 (R) and the real Hardy space H 1 over the real line was pioneered by Liflyand and Móricz in [1]. recently, many research papers have addressed the boundedness of the Hausdorff operator on Hardy spaces. For instance, Liflyand and his collaborators in [2,3] proved, by more effective ways, that the Hausdorff operator has the same behavior on the Hardy space H 1 (R) as that in the Lebesgue space L 1 (R). Dunkl theory generalizes classical Fourier analysis on Rd . It started twenty years ago with Dunkl’s seminal work in [4] and was further developed by several mathematicians. K.Triméche in [5] introduced continuous Dunkl wavelet
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Faouaz Saadi [email protected] Radouan Daher [email protected]
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Department of Mathematics, Laboratory of T.A.G.D.M, Faculty of Sciences, Ain Chock University Hassan II, Casablanca, Morocco
R. Daher, F. Saadi
transformations and studied their properties by exploiting Dunkl convolution. The main purpose of this paper is extending some results of the classical Hausdorff operator given in [6] to the context of Dunkl theory, investigate continuous Dunkl wavelet transformation and obtain some useful results of Dunkl wavelet transformation using the theory of Dunkl convolution. We prove the boundness of Dunkl-Hausdorff operp ator on L α (R) space and Hardy space Hα1 (R). The relation between Dunkl wavelet transformation and Dunkl-Hausdorff operator is also established. To make this paper self-contained, the following results are useful; hence, they are presented in some detail. p
Notation We denote by L α (R), 1 ≤ p ≤ ∞, the space of measurable functions on R such that 1/ p f L αp := | f (x)| p dμα (x) < ∞, if 1 ≤ p < ∞, R
f L ∞ := ess sup | f (x)| < ∞, α x∈R
and μα the measure on R, given by dμα (x) = |x|2α+1 d x. The Dunkl-Hausdorff operator Hα acting on L 1α (R) generated by a function ϕ belonging to L 1 (R) is defined in [7] by: x ϕ(t) dt, f ∈ L 1α (R), x ∈ R. Hα,ϕ f (x) := f 2α+2 t R |t| Making use of Fubini’s theorem yields the absolute convergence of the integral Hα,ϕ f (x) for almost all x ∈ R ( see Pr
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