Lipschitz Conditions in Laguerre Hypergroup

Lipschitz Conditions in Laguerre Hypergroup 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 $$L^p(\mathbb {K})$$ L p ( K ) for which its Laguerre transform belongs to $$L^\beta (\hat{\mathbb {K}})$$ L β ( K ^ ) for some values of $$\beta $$ β , where $$\mathbb {K}=[0,+\infty )\times \mathbb {R}$$ K = [ 0 , + ∞ ) × R and $$\hat{\mathbb {K}}$$ K ^ is its dual. In the particular case, when $$p=2$$ p = 2 , we provide equivalence theorem : we get a characterization of the space $$\mathrm{Lip}_\alpha (\gamma ,2)$$ Lip α ( γ , 2 ) of Lipschitz class functions by means of asymptotic estimate growth of the norm of their Laguerre transform for $$0<\gamma <1$$ 0 < γ < 1 . Furthermore, we introduce Laguerre–Dini–Lipschitz class $$\mathrm{LDLip}_\alpha (\gamma ,\delta ,p)$$ LDLip α ( γ , δ , p ) and we obtain analogous of Titchmarsh’s theorems in this occurence. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mediterranean Journal of Mathematics Springer Journals

Lipschitz Conditions in Laguerre Hypergroup

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Publisher
Springer International Publishing
Copyright
Copyright © 2017 by Springer International Publishing AG
Subject
Mathematics; Mathematics, general
ISSN
1660-5446
eISSN
1660-5454
D.O.I.
10.1007/s00009-017-0989-4
Publisher site
See Article on Publisher Site

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 $$L^p(\mathbb {K})$$ L p ( K ) for which its Laguerre transform belongs to $$L^\beta (\hat{\mathbb {K}})$$ L β ( K ^ ) for some values of $$\beta $$ β , where $$\mathbb {K}=[0,+\infty )\times \mathbb {R}$$ K = [ 0 , + ∞ ) × R and $$\hat{\mathbb {K}}$$ K ^ is its dual. In the particular case, when $$p=2$$ p = 2 , we provide equivalence theorem : we get a characterization of the space $$\mathrm{Lip}_\alpha (\gamma ,2)$$ Lip α ( γ , 2 ) of Lipschitz class functions by means of asymptotic estimate growth of the norm of their Laguerre transform for $$0<\gamma <1$$ 0 < γ < 1 . Furthermore, we introduce Laguerre–Dini–Lipschitz class $$\mathrm{LDLip}_\alpha (\gamma ,\delta ,p)$$ LDLip α ( γ , δ , p ) and we obtain analogous of Titchmarsh’s theorems in this occurence.

Journal

Mediterranean Journal of MathematicsSpringer Journals

Published: Aug 16, 2017

References

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