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Littlewood–Paley–Stein Functions for Hodge-de Rham and Schrödinger Operators

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Abstract

We study the Littlewood–Paley–Stein functions associated with Hodge-de Rham and Schrödinger operators on Riemannian manifolds. Under conditions on the Ricci curvature, we prove their boundedness on \(L^p\) for p in some interval \((p_1,2]\) and make a link to the Riesz Transform. An important fact is that we do not make assumptions of doubling measure or estimates on the heat kernel in this case. For \(p > 2\), we give a criterion to obtain the boundedness of the vertical Littlewood–Paley–Stein function associated with Schrödinger operators on \(L^p\).

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Acknowledgements

The author would like to thank his PhD supervisor El Maati Ouhabaz for his help and advice during the preparation of this article. This research is partly supported by the ANR project RAGE ”Analyse Réelle et Géométrie” (ANR-18-CE40-0012).

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  1. Institut de Mathématiques de Bordeaux, UMR 5251, Université de Bordeaux, 351, Cours de la Libération, 33405, Talence, France

    Thomas Cometx

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Cometx, T. Littlewood–Paley–Stein Functions for Hodge-de Rham and Schrödinger Operators. J Geom Anal 31, 7568–7594 (2021). https://doi.org/10.1007/s12220-020-00569-x

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