On compactness of commutator of multiplication and pseudodifferential operator

Abstract

In this short note we show results on the compactness of the commutator of pseudodifferential operator and operator of multiplication in both \(\mathrm{L}^2\) and \(\mathrm{L}^p\) setting. Our results use the boundedness results of pseudodifferential operators by Hwang and Hwang-Lee, and the Krasnoselskij type interpolation lemma. We use the obtained results to construct a variant of microlocal defect functionals via pseudodifferential operators and derive its localisation principle.

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Acknowledgements

The authors are grateful to the anonymous referee for his/her useful comments.

This work was supported in part by the Croatian Science Foundation under project 9780 WeConMApp, and by the project number 01–417 Advection–diffusion equations in highly heterogeneous media of the Montenegrin Ministry of Science. This work was initialised while the second author was visiting University of Zagreb in the framework of the Marie Curie FP7-PEOPLE-2011-COFUND project Micro-local defect functionals and applications.

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Correspondence to Marin Mišur.

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Dedicated to the memory of Todor V. Gramchev.

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Mišur, M., Mitrović, D. On compactness of commutator of multiplication and pseudodifferential operator. J. Pseudo-Differ. Oper. Appl. 10, 121–131 (2019). https://doi.org/10.1007/s11868-018-0239-y

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Keywords

  • Commutator
  • Compactness
  • Pseudodifferential operator

Mathematics Subject Classification

  • 35S05
  • 47G30