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Spectral Multipliers for the Kohn Laplacian on Forms on the Sphere in \(\mathbb {C}^n\)

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Abstract

The unit sphere \(\mathbb {S}\) in \(\mathbb {C}^n\) is equipped with the tangential Cauchy–Riemann complex and the associated Laplacian \(\Box _b\). We prove a Hörmander spectral multiplier theorem for \(\Box _b\) with critical index \(n-1/2\), that is, half the topological dimension of \(\mathbb {S}\). Our proof is mainly based on representation theory and on a detailed analysis of the spaces of differential forms on \(\mathbb {S}\).

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Acknowledgements

Valentina Casarino and Alessio Martini were partially supported by GNAMPA (Progetto 2014 “Moltiplicatori e proiettori spettrali associati a Laplaciani su sfere e gruppi nilpotenti”) and MIUR (PRIN 2010-2011 “Varietà reali e complesse: geometria, topologia e analisi armonica”). Michael G. Cowling, Alessio Martini, and Adam Sikora were supported by the Australian Research Council (Project DP110102488). Michael G. Cowling and Alessio Martini acknowledge the generous support of the Humboldt Foundation. Alessio Martini gratefully acknowledges the support of Deutsche Forschungsgemeinschaft (project MA 5222/2-1).

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Correspondence to Michael G. Cowling.

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Casarino, V., Cowling, M.G., Martini, A. et al. Spectral Multipliers for the Kohn Laplacian on Forms on the Sphere in \(\mathbb {C}^n\) . J Geom Anal 27, 3302–3338 (2017). https://doi.org/10.1007/s12220-017-9806-3

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