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Asymptotic Estimates for the Growth of Deformed Hankel Transform by Modulus of Continuity

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Abstract

We derive asymptotic estimates for the growth of the norm of the deformed Hankel transform on the deformed Hankel–Lipschitz space defined via a generalised modulus of continuity. The established results are similar in nature to the well-known Titchmarsh theorem, which provide a characterization of the square integrable functions satisfying certain Cauchy–Lipschitz condition in terms of an asymptotic estimate for the growth of the norm of their Fourier transform. We also give some necessary conditions in terms of the generalised modulus of continuity for the boundedness of the Dunkl transform of functions in Dunkl–Lipschitz spaces, improving the Hausdorff–Young inequality for the Dunkl transform in this special scenario.

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Acknowledgements

The authors were supported by the FWO Odysseus 1 grant G.0H94.18N: Analysis and Partial Differential Equations, the Methusalem programme of the Ghent University Special Research Fund (BOF) (Grant number 01M01021) and by FWO Senior Research Grant G011522N. MR is also supported by EPSRC grant EP/R003025/2.

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Authors and Affiliations

  1. Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Krijgslaan 281, Building S8, 9000, Ghent, Belgium

    Vishvesh Kumar, Joel E. Restrepo & Michael Ruzhansky

  2. School of Mathematical Sciences, Queen Mary University of London, London, United Kingdom

    Michael Ruzhansky

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  1. Vishvesh Kumar
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  2. Joel E. Restrepo
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  3. Michael Ruzhansky
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Correspondence to Joel E. Restrepo.

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Kumar, V., Restrepo, J.E. & Ruzhansky, M. Asymptotic Estimates for the Growth of Deformed Hankel Transform by Modulus of Continuity. Results Math 79, 22 (2024). https://doi.org/10.1007/s00025-023-02051-w

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