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Basis Properties of the Haar System in Limiting Besov Spaces

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Geometric Aspects of Harmonic Analysis

Part of the book series: Springer INdAM Series ((SINDAMS,volume 45))

Abstract

We study Schauder basis properties for the Haar system in Besov spaces \(B^s_{p,q}(\mathbb {R}^d)\). We give a complete description of the limiting cases, obtaining various positive results for \(q\leq \min \{1,p\}\), and providing new counterexamples in other situations. The study is based on suitable estimates of the dyadic averaging operators \(\mathbb {E}_N\); in particular we find asymptotically optimal growth rates for the norms of these operators in global and local situations.

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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 DMS 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, Murcia, Spain

    Gustavo Garrigós

  2. Department of Mathematics, University of Wisconsin, Madison, WI, USA

    Andreas Seeger

  3. Faculty of Mathematics, Technical University of Chemnitz, Chemnitz, Germany

    Tino Ullrich

Authors
  1. Gustavo Garrigós
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  2. Andreas Seeger
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  3. Tino Ullrich
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Corresponding author

Correspondence to Andreas Seeger .

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Editors and Affiliations

  1. Dipartimento di Ingegneria Civile, Edile e Ambientale - ICEA, University of Padova, Padova, Italy

    Paolo Ciatti

  2. School of Mathematics, University of Birmingham, Birmingham, UK

    Alessio Martini

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Garrigós, G., Seeger, A., Ullrich, T. (2021). Basis Properties of the Haar System in Limiting Besov Spaces. In: Ciatti, P., Martini, A. (eds) Geometric Aspects of Harmonic Analysis. Springer INdAM Series, vol 45. Springer, Cham. https://doi.org/10.1007/978-3-030-72058-2_11

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