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Spectral Invariance of Non-Smooth Pseudo-Differential Operators

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Abstract

We discuss some spectral invariance results for non-smooth pseudodifferential operators with coefficients in Hölder spaces in this paper. In analogy to the proof in the smooth case of Beals and Ueberberg, c.f. (Duke Math J 44(1):45–57, 1977; Manuscripta Math 61(4):459–475, 1988), we use the characterization of non-smooth pseudodifferential operators to get such a result. The main new difficulties are the limited mapping properties of pseudodifferential operators with non-smooth symbols and the fact, that in general the composition of two non-smooth pseudodifferential operators is not a pseudodifferential operator. In order to improve these spectral invariance results for certain subsets of non-smooth pseudodifferential operators with coefficients in Hölder spaces, we improve the characterization of non-smooth pseudodifferential operators of A. and P., c.f. (Abels and Pfeuffer, Characterization of non-smooth pseudodifferential operators. arXiv:1512.01127, 2015).

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Authors and Affiliations

  1. Universität Regensburg, Universitätsstraße 31, 93053, Regensburg, Germany

    Helmut Abels & Christine Pfeuffer

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  1. Helmut Abels
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  2. Christine Pfeuffer
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Correspondence to Helmut Abels.

Additional information

We would like to thank Prof. Dr. Schrohe for his helpful suggestions of improvements. Moreover, we thank the referee for the careful reading of the manuscript and many valuable suggestions for improvements.

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Abels, H., Pfeuffer, C. Spectral Invariance of Non-Smooth Pseudo-Differential Operators. Integr. Equ. Oper. Theory 86, 41–70 (2016). https://doi.org/10.1007/s00020-016-2315-0

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  • DOI: https://doi.org/10.1007/s00020-016-2315-0

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