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The Haar System in Triebel–Lizorkin Spaces: Endpoint Results

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Abstract

We characterize the Schauder and the unconditional basis properties for the Haar system in the Triebel–Lizorkin spaces \(F^s_{p,q}({{\mathbb {R}}}^d)\), at the endpoint cases \(s=1\), \(s=d/p-d\), and \(p=\infty \). Together with the earlier results in Garrigós et al. (J Fourier Anal Appl 24(5):1319–1339, 2018) and Seeger and Ullrich (Math Z 285:91–119, 2017), this completes the picture for such properties in the Triebel–Lizorkin scale, and complements a similar study for the Besov spaces given in Garrigós et al. (Basis properties of the Haar system in limiting Besov spaces. In: Geometric aspects of harmonic analysis: a conference in honour of Fulvio Ricci, Springer-INdAM series, arXiv.1901.09117).

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Notes

  1. In the notation of [10, §6], one should consider sets A of consecutive Haar frequencies, so that the associated “density” number in [10, (43)] takes the value \(Z=N\).

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Acknowledgements

The authors would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the Program Approximation, Sampling and Compression in Data Science where some work on this paper was undertaken. This work was supported by EPSRC Grant No. EP/K032208/1. G.G. was supported in part by Grants MTM2016-76566-P, MTM2017-83262-C2-2-P and Programa Salvador de Madariaga PRX18/451 from Micinn (Spain), and Grant 20906/PI/18 from Fundación Séneca (Región de Murcia, Spain). A.S. was supported in part by National Science Foundation Grants 1500162 and 1764295. T.U. was supported in part by Deutsche Forschungsgemeinschaft (DFG), Grant 403/2-1.

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Authors and Affiliations

  1. Department of Mathematics, University of Murcia, Espinardo, 30100, Murcia, Spain

    Gustavo Garrigós

  2. Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, WI, 53706, USA

    Andreas Seeger

  3. Fakultät für Mathematik, Technische Universität Chemnitz, 09107, Chemnitz, Germany

    Tino Ullrich

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  1. Gustavo Garrigós
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  2. Andreas Seeger
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  3. Tino Ullrich
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Correspondence to Gustavo Garrigós.

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Dedicated to Guido Weiss, with affection, on his 92nd birthday.

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Garrigós, G., Seeger, A. & Ullrich, T. The Haar System in Triebel–Lizorkin Spaces: Endpoint Results. J Geom Anal 31, 9045–9089 (2021). https://doi.org/10.1007/s12220-020-00577-x

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