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Hörmander functional calculus on UMD lattice valued Lp spaces under generalized Gaussian estimates

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Abstract

We consider self-adjoint semigroups Tt = exp(−tA) acting on L2(Ω) and satisfying (generalized) Gaussian estimates, where Ω is a metric measure space of homogeneous type of dimension d. The aim of the article is to show that A ⨂ IdY admits a Hörmander type \({\cal H}_2^\beta \) functional calculus on Lp (Ω; Y) where Y is a UMD lattice, thus extending the well-known Hörmander calculus of A on Lp (Ω). We show that if Tt is lattice positive (or merely admits an H∞ calculus on Lp (Ω; Y)) then this is indeed the case. Here the derivation exponent has to satisfy \(\beta > \alpha \cdot d + {1 \over 2}\), where α ∈ (0, 1) depends on p, and on convexity and concavity exponents of Y. A part of the proof is the new result that the Hardy-Littlewood maximal operator is bounded on Lp(Ω; Y). Moreover, our spectral multipliers satisfy square function estimates in Lp(Ω; Y). In a variant, we show that if eitA satisfies a dispersive L1(Ω) → L∞(Ω) estimate, then \(\beta > {{d + 1} \over 2}\) above is admissible independent of convexity and concavity of Y. Finally, we illustrate these results in a variety of examples.

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Acknowledgments

The first and third author are financially supported by the grant ANR-18-CE40-0021 of the French National Research Agency ANR (project HASCON). The first author is financially supported by the grant ANR-18-CE40-0035 (project REPKA) and the third author is financially supported by the grant ANR-17-CE40-0021 (project Front).

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  1. Laboratoire d’Analyse et de Mathématiques Appliquées (UMR 8050), Université Gustave Eiffel, 5, Boulevard Descartes, Champs sur Marne, 77454, Marne la Vallée Cedex 2, France

    Luc Deleaval

  2. Limingantie 73 as 19, 00560, Helsinki, Finland

    Mikko Kemppainen

  3. Laboratoire de Mathématiques Blaise Pascal (CNRS UMR 6620), Université Clermont Auvergne, 63 000, Clermont-Ferrand, France

    Christoph Kriegler

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  1. Luc Deleaval
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Deleaval, L., Kemppainen, M. & Kriegler, C. Hörmander functional calculus on UMD lattice valued Lp spaces under generalized Gaussian estimates. JAMA 145, 177–234 (2021). https://doi.org/10.1007/s11854-021-0177-0

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