Abstract
The purpose of this paper is to prove analogous of Titchmarsh’s theorems for the Laguerre transform. More precisely, we give a Lipschitz-type condition on f in \(L^p(\mathbb {K})\) for which its Laguerre transform belongs to \(L^\beta (\hat{\mathbb {K}})\) for some values of \(\beta \), where \(\mathbb {K}=[0,+\infty )\times \mathbb {R}\) and \(\hat{\mathbb {K}}\) is its dual. In the particular case, when \(p=2\), we provide equivalence theorem : we get a characterization of the space \(\mathrm{Lip}_\alpha (\gamma ,2)\) of Lipschitz class functions by means of asymptotic estimate growth of the norm of their Laguerre transform for \(0<\gamma <1\). Furthermore, we introduce Laguerre–Dini–Lipschitz class \(\mathrm{LDLip}_\alpha (\gamma ,\delta ,p)\) and we obtain analogous of Titchmarsh’s theorems in this occurence.
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Negzaoui, S. Lipschitz Conditions in Laguerre Hypergroup. Mediterr. J. Math. 14, 191 (2017). https://doi.org/10.1007/s00009-017-0989-4
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DOI: https://doi.org/10.1007/s00009-017-0989-4