Abstract
It was shown in [Colloq. Math. 131(2), 219--231 (2013)] that one can extend the domain of Fourier transform of a commutative hypergroup K to \(L^p(K)\) for \(1\le p \le 2\), and the Hausdorff–Young inequality holds true for these cases. In this article, we examine the structure of non-zero functions in \(L^p(K)\) for which equality is attained in the Hausdorff–Young inequality, for \(1<p<2\), and further provide a characterization for the basic uncertainty principle for commutative hypergroups with non-trivial center.
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References
Babenko, K.I: An inequality in the theory of Fourier integrals. Izv. Akad. Nauk SSSR Ser. Mat. 25, 531–542 (1961) ((Russian) translated as Am. Math. Soc. Transl. Ser. 2 44:115-128 (1962))
Bandyopadhyay, C.: Analysis on semihypergroups: function spaces, homomorphisms and ideals. Semigroup Forum 100(3), 671–697 (2020)
Bandyopadhyay, C.: Free product of semihypergroups. Semigroup Forum 102(1), 28–47 (2021)
Beckner, W.: Inequalities in Fourier analysis. Ann. Math. 2(102), 159–182 (1975)
Bloom, W.R., Heyer, H.: Harmonic Analysis of Probability Measures on Hypergroups. De Gruyter Studies in Mathematics, vol. 20. Walter de Gruyter & Co., Berlin (1995)
Cowling, M.G., Martini, A., Müller, D., Parcet, J.: The Hausdorff–Young inequality on Lie groups. Math. Ann. 375(1–2), 93–131 (2019)
Degenfeld-Schonburg, S.: On the Hausdorff–Young theorem for commutative hypergroups. Colloq. Math. 131(2), 219–231 (2013)
Hewitt, E., Hirschman, I., Jr.: A maximum problem in harmonic analysis. Am. J. Math. 76, 839–852 (1954)
Hewitt, E., Ross, K.A.: Abstract Harmonic Analysis. Vol. II: Structure and Analysis for Compact Groups. Analysis on Locally Compact Abelian Groups. Die Grundlehren der mathematischen Wissenschaften, Band, vol. 152. Springer, New York (1970)
Hirschman, I., Jr.: A maximal problem in harmonic analysis II. Pac. J. Math. 9, 525–540 (1959)
Jewett, R.I.: Spaces with an abstract convolution of measures. Adv. Math. 18(1), 1–101 (1975)
Landstad, M.B., Van Daele, A.: Groups with compact open subgroups and multiplier Hopf \(*\)-algebras. Expo. Math. 26(3), 197–217 (2008)
Michael, E.: Topologies on spaces of subsets. Trans. Am. Math. Soc. 71, 152–182 (1951)
Ross, K.A.: Centres of hypergroups. Trans. Am. Math. Soc. 243, 251–269 (1978)
Russo, B.: The norm of the \(L^p\)-Fourier transform on unimodular groups. Trans. Am. Math. Soc. 192, 293–305 (1974)
Takai, H.: On a duality for crossed products of C*-algebras. J. Funct. Anal. 19, 25–39 (1975)
Acknowledgements
The authors are grateful to the referee for the thoughtful remarks and suggestions, which led to overall improvement of the article. The first named author would like to gratefully acknowledge the financial support provided by the Indian Institute of Technology Kanpur, India, throughout the course of this research.
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Communicated by Yong Zhang.
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Bandyopadhyay, C., Mohanty, P. Equality in Hausdorff–Young for hypergroups. Banach J. Math. Anal. 16, 62 (2022). https://doi.org/10.1007/s43037-022-00211-8
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DOI: https://doi.org/10.1007/s43037-022-00211-8