General Coorbit Space Theory for Quasi-Banach Spaces and Inhomogeneous Function Spaces with Variable Smoothness and Integrability

Journal of Fourier Analysis and Applications volume 23, pages 1348–1407 (2017)Cite this article

Abstract

In this paper we propose a general coorbit space theory suitable to define coorbits of quasi-Banach spaces using an abstract continuous frame, indexed by a locally compact Hausdorff space, and an associated generalized voice transform. The proposed theory realizes a further step in the development of a universal abstract theory towards various function spaces and their atomic decompositions which has been initiated by Feichtinger and Gröchenig in the late 1980s. We combine the recent approaches in Rauhut and Ullrich (J Funct Anal 260(11):3299–3362, 2011) and Rauhut (Stud Math 180(3):237–253, 2007) to identify, in particular, various inhomogeneous (quasi-Banach) spaces of Besov–Lizorkin–Triebel type. To prove the potential of our new theory we apply it to spaces with variable smoothness and integrability which have attracted significant interest in the last 10 years. From the abstract discretization machinery we obtain atomic decompositions as well as wavelet frame isomorphisms for these spaces.

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Acknowledgments

The authors would like to thank Felix Voigtlaender for a careful reading of the manuscript and many valuable comments and corrections. They would further like to thank the anonymous referees for their careful proofreading and many valuable remarks. In particular, Lemma 2.14 should be pointed out, where now—based on their input—also questions of Bochner-measurability are discussed. Furthermore, a serious technical issue in the proof of Theorem 3.11 has been fixed. Tino Ullrich gratefully acknowledges support by the German Research Foundation (DFG) Ul-403/2-1 as well as the Emmy-Noether programme, Ul-403/1-1. Martin Schäfer acknowledges support by the Berlin Mathematical School. Further, he would like to thank Holger Rauhut for support during his diploma studies where some ideas of this paper were developed. Henning Kempka acknowledges the support by the German Research Foundation (DFG) within the project Ke 1847/1-1.

Author information

Affiliations

  1. Faculty of Mathematics, Technical University Chemnitz, Reichenhainer Straße 39, 09126, Chemnitz, Germany

    Henning Kempka

  2. Institute of Mathematics, Technische Universität Berlin, Straße des 17. Juni 136, 10623, Berlin, Germany

    Martin Schäfer

  3. Hausdorff Center for Mathematics & Institute for Numerical Simulation, Endenicher Allee 62, 53115, Bonn, Germany

    Tino Ullrich

Authors
  1. Henning Kempka
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  2. Martin Schäfer
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  3. Tino Ullrich
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Correspondence to Henning Kempka.

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Communicated by Stephan Dahlke.

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Kempka, H., Schäfer, M. & Ullrich, T. General Coorbit Space Theory for Quasi-Banach Spaces and Inhomogeneous Function Spaces with Variable Smoothness and Integrability. J Fourier Anal Appl 23, 1348–1407 (2017). https://doi.org/10.1007/s00041-016-9505-7

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Keywords

  • Coorbit space theory
  • Continuous wavelet transform
  • Besov–Lizorkin–Triebel type spaces
  • Variable smoothness
  • Variable integrability
  • 2-Microlocal spaces
  • Peetre maximal function
  • Atomic decomposition
  • Wavelet bases

Mathematics Subject Classification

  • 42B25
  • 42B35
  • 46E35
  • 46F05