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Conical square function estimates in UMD Banach spaces and applications to H ∞-functional calculi

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Abstract

We study conical square function estimates for Banach-valued functions and introduce a vector-valued analogue of the Coifman-Meyer-Stein tent spaces. Following recent work of Auscher-McIntosh-Russ, the tent spaces in turn are used to construct a scale of vector-valued Hardy spaces associated with a given bisectorial operator A with certain off-diagonal bounds such that A always has a bounded H ∞-functional calculus on these spaces. This provides a new way of proving functional calculus of A on the Bochner spaces L p(ℝn; X) by checking appropriate conical square function estimates and also a conical analogue ofBourgain’s extension of the Littlewood-Paley theory to the UMD-valued context. Even when X = ℂ, our approach gives refined p-dependent versions of known results.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Helsinki, Gustaf Hällströmin katu 2b, FI-00014, Helsinki, Finland

    Tuomas Hytönen

  2. Delft Institute of Applied Mathematics, Delft University of Technology, P.O. Box 5031, 2600 GA, Delft, The Netherlands

    Jan van Neerven

  3. Mathematical Sciences Institute, Australian National University, ACT, Building 27, 0200, Canberra, Australia

    Pierre Portal

Authors
  1. Tuomas Hytönen
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  2. Jan van Neerven
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  3. Pierre Portal
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Correspondence to Tuomas Hytönen.

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Hytönen, T., van Neerven, J. & Portal, P. Conical square function estimates in UMD Banach spaces and applications to H ∞-functional calculi. J Anal Math 106, 317–351 (2008). https://doi.org/10.1007/s11854-008-0051-3

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  • DOI: https://doi.org/10.1007/s11854-008-0051-3

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