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Haar projection numbers and failure of unconditional convergence in Sobolev spaces

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Abstract

For \(1<p<\infty \) we determine the precise range of \(L_p\) Sobolev spaces for which the Haar system is an unconditional basis. We also consider the natural extensions to Triebel–Lizorkin spaces and prove upper and lower bounds for norms of projection operators depending on properties of the Haar frequency set.

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Authors and Affiliations

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

    Andreas Seeger

  2. Hausdorff Center for Mathematics, Endenicher Allee 62, 53115, Bonn, Germany

    Tino Ullrich

Authors
  1. Andreas Seeger
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  2. Tino Ullrich
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Correspondence to Andreas Seeger.

Additional information

Research supported in part by National Science Foundation grants DMS 1200261 and DMS 1500162. A.S. thanks the Hausdorff Research Institute for Mathematics in Bonn for support. The paper was initiated in the summer of 2014 when he participated in the Hausdorff Trimester Program in Harmonic Analysis and Partial Differential Equations. Both authors would like to thank Winfried Sickel and Hans Triebel for several valuable remarks.

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Seeger, A., Ullrich, T. Haar projection numbers and failure of unconditional convergence in Sobolev spaces. Math. Z. 285, 91–119 (2017). https://doi.org/10.1007/s00209-016-1697-7

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  • DOI: https://doi.org/10.1007/s00209-016-1697-7

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