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$\boldsymbol {L}^{\boldsymbol {p}}$–$\boldsymbol {L}^{\boldsymbol {q}}$ MULTIPLIERS ON COMMUTATIVE HYPERGROUPS

Part of: Harmonic analysis in one variable Abstract harmonic analysis Harmonic analysis in several variables

Published online by Cambridge University Press:  18 October 2023

VISHVESH KUMAR Open the ORCID record for VISHVESH KUMAR [Opens in a new window]  and
MICHAEL RUZHANSKY Open the ORCID record for MICHAEL RUZHANSKY [Opens in a new window]
VISHVESH KUMAR*
Affiliation:
Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Ghent, Belgium
MICHAEL RUZHANSKY
Affiliation:
Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Ghent, Belgium and School of Mathematical Sciences, Queen Mary University of London, London, UK e-mail: michael.ruzhansky@ugent.be
*
e-mail: vishveshmishra@gmail.com
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Abstract

The main purpose of this paper is to prove Hörmander’s $L^p$–$L^q$ boundedness of Fourier multipliers on commutative hypergroups. We carry out this objective by establishing the Paley inequality and Hausdorff–Young–Paley inequality for commutative hypergroups. We show the $L^p$–$L^q$ boundedness of the spectral multipliers for the generalised radial Laplacian by examining our results on Chébli–Trimèche hypergroups. As a consequence, we obtain embedding theorems and time asymptotics for the $L^p$–$L^q$ norms of the heat kernel for generalised radial Laplacian.

Keywords

Chébli–Trimèche hypergroupsFourier multipliersspectral multipliersJacobi transformHankel transformgeneralised radial Laplacian

MSC classification

Primary: 43A62: Hypergroups
Secondary: 42B10: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type 42A45: Multipliers
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 115 , Issue 3 , December 2023 , pp. 375 - 395
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by Ji Li

The authors are supported by 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 the EPSRC Grant EP/R003025/2 and by the FWO Grant G022821N.

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